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A partial differential equation which is of varying type (elliptic, hyperbolic or parabolic) in its domain of definition. A linear (or quasi-linear) differential equation of the second order with two unknown variables,
 
A partial differential equation which is of varying type (elliptic, hyperbolic or parabolic) in its domain of definition. A linear (or quasi-linear) differential equation of the second order with two unknown variables,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
A u _ {xx} + 2 B u _ {xy} + C u _ {yy}  = \
 +
f ( x , y , u , u _ {x} , u _ {y} )
 +
$$
  
and with coefficients defined in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642502.png" /> is an equation of mixed type if the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642503.png" /> of the characteristic form
+
and with coefficients defined in the domain $  \Omega $
 +
is an equation of mixed type if the discriminant $  \Delta = A C - B  ^ {2} $
 +
of the characteristic form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642504.png" /></td> </tr></table>
+
$$
 +
A  d y  ^ {2} + 2 B  d x  d y + C  d x  ^ {2}  = Q
 +
$$
  
takes the value zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642505.png" /> but is not identically zero there.
+
takes the value zero in $  \Omega $
 +
but is not identically zero there.
  
The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642506.png" /> defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642507.png" /> is called the parabolic line of equation (1) or the line of degeneracy (change) of type.
+
The curve $  \delta $
 +
defined by the equation $  \Delta = 0 $
 +
is called the parabolic line of equation (1) or the line of degeneracy (change) of type.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642508.png" /> does not change sign when the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m0642509.png" /> crosses the parabolic line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425011.png" />, then equation (1) is a degenerate equation of elliptic-parabolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425012.png" /> or hyperbolic-parabolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425013.png" /> type (see [[Degenerate partial differential equation|Degenerate partial differential equation]]).
+
If $  \Delta $
 +
does not change sign when the point $  ( x , y ) $
 +
crosses the parabolic line $  \delta $
 +
in $  \Omega $,  
 +
then equation (1) is a degenerate equation of elliptic-parabolic $  ( \Delta \geq  0 ) $
 +
or hyperbolic-parabolic $  ( \Delta \leq  0 ) $
 +
type (see [[Degenerate partial differential equation|Degenerate partial differential equation]]).
  
Under certain smoothness conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425017.png" />, there is a non-singular real transformation of the independent variables sending equation (1) (in the case where the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425018.png" /> alternates in the neighbourhood of a chosen point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425019.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425020.png" />) to one of the following canonical forms (the notation for the independent variables is preserved):
+
Under certain smoothness conditions on $  A $,  
 +
$  B $,  
 +
$  C $,  
 +
and $  \delta $,  
 +
there is a non-singular real transformation of the independent variables sending equation (1) (in the case where the sign of $  \Delta $
 +
alternates in the neighbourhood of a chosen point of $  \delta $
 +
where $  A  ^ {2} + B  ^ {2} \neq 0 $)  
 +
to one of the following canonical forms (the notation for the independent variables is preserved):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
y  ^ {2m+1} u _ {xx} + u _ {yy}  = \
 +
F ( x , y , u , u _ {x} , u _ {y} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
u _ {xx} + y  ^ {2m+1} u _ {yy}  = F ( x , y , u , u _ {x} , u _ {y} ) .
 +
$$
  
The equations (2) and (3) are equations of mixed (elliptic-hyperbolic) type in any domain containing a segment of the line of degeneracy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425023.png" />.
+
The equations (2) and (3) are equations of mixed (elliptic-hyperbolic) type in any domain containing a segment of the line of degeneracy $  y = 0 $.
  
The domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425024.png" /> of definition of an equation of mixed type is sometimes called a mixed domain, and boundary value problems in mixed domains are called mixed boundary value problems. The part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425026.png" />) of a mixed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425027.png" /> where the equation is of elliptic (hyperbolic) type is called the domain of ellipticity (hyperbolicity).
+
The domain $  \Omega $
 +
of definition of an equation of mixed type is sometimes called a mixed domain, and boundary value problems in mixed domains are called mixed boundary value problems. The part $  \Omega  ^ {+} $ ($  \Omega  ^ {-} $)  
 +
of a mixed domain $  \Omega $
 +
where the equation is of elliptic (hyperbolic) type is called the domain of ellipticity (hyperbolicity).
  
 
Many problems of an applied nature reduce to finding specific solutions of equations of mixed type; in particular, problems of plane transonic flow of a compressible medium, and problems in the theory of envelopes.
 
Many problems of an applied nature reduce to finding specific solutions of equations of mixed type; in particular, problems of plane transonic flow of a compressible medium, and problems in the theory of envelopes.
  
An equation (1) of mixed type is called an equation of the first kind (second kind) if the characteristic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425028.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425029.png" />) everywhere on the parabolic line. The Chaplygin equation
+
An equation (1) of mixed type is called an equation of the first kind (second kind) if the characteristic form $  Q \neq 0 $ ($  Q = 0 $)  
 +
everywhere on the parabolic line. The Chaplygin equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
k ( y) u _ {xx} + u _ {yy}  = 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425031.png" /> is a continuously-differentiable monotone function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425032.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425033.png" />, is a typical example of an equation of mixed type of the first kind. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425034.png" />, equation (4) is usually called the [[Tricomi equation|Tricomi equation]].
+
where $  k ( y) $
 +
is a continuously-differentiable monotone function such that $  y k ( y) > 0 $
 +
when $  y \neq 0 $,  
 +
is a typical example of an equation of mixed type of the first kind. When $  k ( y) = y $,  
 +
equation (4) is usually called the [[Tricomi equation|Tricomi equation]].
  
 
An important model for an equation of mixed type (with a discontinuous coefficient in front of one of the higher derivatives) is the equation of Lavrent'ev–Bitsadze
 
An important model for an equation of mixed type (with a discontinuous coefficient in front of one of the higher derivatives) is the equation of Lavrent'ev–Bitsadze
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
(  \mathop{\rm sign}  y ) \cdot u _ {xx} + u _ {yy}  = 0 .
 +
$$
  
One of the basic boundary value problems for equations of mixed type (of the first kind) is the Tricomi problem, which is as follows for an equation of the form (2). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425036.png" /> be a finite simply-connected domain in the Euclidean plane with independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425038.png" />, bounded by a simple Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425039.png" /> with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425041.png" /> lying in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425042.png" />, and by the parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425044.png" /> of the characteristics of equation (2) going through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425046.png" />. The Tricomi problem is to find a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425047.png" /> of equation (2) that is continuous in the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425049.png" /> and takes given values on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425050.png" />.
+
One of the basic boundary value problems for equations of mixed type (of the first kind) is the Tricomi problem, which is as follows for an equation of the form (2). Let $  \Omega $
 +
be a finite simply-connected domain in the Euclidean plane with independent variables $  x $
 +
and $  y $,  
 +
bounded by a simple Jordan curve $  \sigma $
 +
with end points $  A ( 0 , 0 ) $,
 +
$  B ( 1 , 0 ) $
 +
lying in the half-plane $  y > 0 $,  
 +
and by the parts $  A C $
 +
and $  B C $
 +
of the characteristics of equation (2) going through the point $  C ( 1/2 , y _ {C} ) $,  
 +
$  y _ {C} < 0 $.  
 +
The Tricomi problem is to find a solution $  u ( x , y ) $
 +
of equation (2) that is continuous in the closure $  \overline \Omega \; $
 +
of $  \Omega $
 +
and takes given values on the curve $  \sigma \cup A C $.
  
In the theory of the Tricomi problem, an essential role is played by the Bitsadze extremum principle, which, in the case of equation (5), states that a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425051.png" /> of equation (5) in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425052.png" /> which vanishes on the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425053.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425055.png" />, in the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425056.png" /> of the domain of ellipticity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425057.png" />, attains its extremum on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425058.png" />.
+
In the theory of the Tricomi problem, an essential role is played by the Bitsadze extremum principle, which, in the case of equation (5), states that a solution $  u ( x , y ) $
 +
of equation (5) in the class $  C ( \overline \Omega ) \cap C  ^ {1} ( \Omega ) $
 +
which vanishes on the characteristic $  A C $:  
 +
$  x + y = 0 $,  
 +
0 \leq  x \leq  1/2 $,  
 +
in the closure $  \overline \Omega  ^ {+} $
 +
of the domain of ellipticity $  \Omega  ^ {+} = \Omega \cap \{ y > 0 \} $,  
 +
attains its extremum on the curve $  \sigma $.
  
This principle, which guarantees the uniqueness and stability of a solution of the Tricomi problem (and also provides a basic estimate to prove existence by the alternating method), can be generalized to a very wide class of linear and quasi-linear equations of mixed type. In particular, it applies to Chaplygin equations (and Tricomi equations) whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425059.png" /> is twice continuously differentiable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425060.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425061.png" />. The Bitsadze extremum principle also holds for the equation
+
This principle, which guarantees the uniqueness and stability of a solution of the Tricomi problem (and also provides a basic estimate to prove existence by the alternating method), can be generalized to a very wide class of linear and quasi-linear equations of mixed type. In particular, it applies to Chaplygin equations (and Tricomi equations) whenever $  k ( y) $
 +
is twice continuously differentiable and $  5 k ^ {\prime 2 } \geq  4 k k  ^  \prime  $
 +
for $  y < 0 $.  
 +
The Bitsadze extremum principle also holds for the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
\mathop{\rm sign}  y \cdot | y |  ^  \alpha  u _ {xx} +
 +
u _ {yy}  = 0 , \alpha = \textrm{ const } > 0 .
 +
$$
  
The solution of the Tricomi problem for equation (6) in the corresponding mixed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425063.png" /> can be written in explicit form when the elliptic part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425064.png" /> of the boundary of this domain coincides with the so-called normal contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425065.png" />:
+
The solution of the Tricomi problem for equation (6) in the corresponding mixed domain $  \Omega $
 +
can be written in explicit form when the elliptic part $  \sigma $
 +
of the boundary of this domain coincides with the so-called normal contour $  \sigma _ {0} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425066.png" /></td> </tr></table>
+
$$
 +
x  ^ {2} + \left (
 +
\frac{2}{\alpha + 2 }
 +
\right )  ^ {2}
 +
y ^ {\alpha + 2 }  =
 +
\frac{1}{4}
 +
.
 +
$$
  
In the general case, under specific conditions on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425067.png" /> and on the class in which the solutions are sought, the Tricomi problem for equation (6) reduces to an equivalent [[Singular integral equation|singular integral equation]] which is (by virtue of the uniqueness condition) unconditionally solvable. The method of integral equations can also be applied to prove existence of a solution of the Tricomi problem, and of other mixed problems, for more general equations of the form
+
In the general case, under specific conditions on the curve $  \sigma $
 +
and on the class in which the solutions are sought, the Tricomi problem for equation (6) reduces to an equivalent [[Singular integral equation|singular integral equation]] which is (by virtue of the uniqueness condition) unconditionally solvable. The method of integral equations can also be applied to prove existence of a solution of the Tricomi problem, and of other mixed problems, for more general equations of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425068.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sign}  y \cdot | y |  ^  \alpha
 +
u _ {xx} + u _ {yy}  = F ( x , y , u , u _ {x} , u _ {y} )
 +
$$
  
with power degeneracy of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425069.png" />.
+
with power degeneracy of order $  \alpha $.
  
 
Function-theoretical methods and functional analysis, in particular the use of a priori estimates, have made it possible to extend significantly the class of equations of mixed type and mixed domains for which existence and uniqueness of a (generalized) solution can be proved, both for the Tricomi problems and for various other mixed problems.
 
Function-theoretical methods and functional analysis, in particular the use of a priori estimates, have made it possible to extend significantly the class of equations of mixed type and mixed domains for which existence and uniqueness of a (generalized) solution can be proved, both for the Tricomi problems and for various other mixed problems.
  
An important generalization of the Tricomi problem is the general mixed Bitsadze problem, which, in the case of equation (5), can be posed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425070.png" /> be a simply-connected mixed domain bounded by a simple Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425071.png" /> lying in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425072.png" /> with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425074.png" /> and (smooth) monotone curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425076.png" /> through these points, meeting at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425078.png" />. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425080.png" /> lie in the domain bounded by the characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425082.png" /> and the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425083.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425084.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425085.png" />-axis. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425087.png" /> denote the points of intersection of the characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425089.png" /> with the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425091.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425092.png" /> is any fixed point of the semi-interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425093.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425095.png" /> denote the parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425097.png" /> lying between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425098.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m06425099.png" />, respectively. The general mixed Bitsadze problem consists of finding a regular solution (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250101.png" />) of equation (5) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250102.png" /> which is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250103.png" />, has continuous first derivatives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250104.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250105.png" />, and satisfies given boundary conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250108.png" />. The uniqueness and existence of a solution of this problem, both for equation (5) and for more general equations, can be proved under certain geometric conditions on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250109.png" />, especially on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250110.png" />. The general mixed Bitsadze problem can be regarded as completely solved in the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250111.png" /> coincides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250112.png" /> with the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250113.png" /> through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250114.png" />. An important consequence of the fact that the general mixed Bitsadze problem is correctly posed, in the case of equation (5) for example, is that the [[Dirichlet problem|Dirichlet problem]] for mixed domains of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250115.png" /> is incorrectly posed, whatever the size and form of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250116.png" /> of hyperbolicity.
+
An important generalization of the Tricomi problem is the general mixed Bitsadze problem, which, in the case of equation (5), can be posed as follows. Let $  \Omega $
 +
be a simply-connected mixed domain bounded by a simple Jordan curve $  \sigma $
 +
lying in the half-plane $  y > 0 $
 +
with end points $  A ( 0 , 0 ) $,
 +
$  B ( 1 , 0 ) $
 +
and (smooth) monotone curves $  \Gamma _ {0} $
 +
and $  \Gamma _ {1} $
 +
through these points, meeting at a point $  C ( x _ {1} , y _ {1} ) $,  
 +
$  y _ {1} < 0 $.  
 +
It is assumed that $  \Gamma _ {0} $
 +
and $  \Gamma _ {1} $
 +
lie in the domain bounded by the characteristics $  x + y = 0 $,  
 +
$  x - y = 1 $
 +
and the interval $  A B $:  
 +
0 \leq  x \leq  1 $
 +
on the $  x $-axis. Let $  B _ {0} $
 +
and $  B _ {1} $
 +
denote the points of intersection of the characteristics $  x - y = x _ {0} $
 +
and $  x + y = x _ {0} $
 +
with the curves $  \Gamma _ {0} $
 +
and $  \Gamma _ {1} $,  
 +
where $  x _ {0} $
 +
is any fixed point of the semi-interval $  x _ {1} + y _ {1} < x _ {1} \leq  x _ {1} - y _ {1} $,  
 +
and let $  \gamma _ {0} $
 +
and $  \gamma _ {1} $
 +
denote the parts of $  \Gamma _ {0} $
 +
and $  \Gamma _ {1} $
 +
lying between $  A , B _ {0} $,  
 +
and $  B , B _ {1} $,  
 +
respectively. The general mixed Bitsadze problem consists of finding a regular solution (when $  y \neq 0 $,  
 +
$  x \pm  y \neq x _ {0} $)  
 +
of equation (5) in $  \Omega $
 +
which is continuous in $  \overline \Omega \; $,  
 +
has continuous first derivatives in $  \Omega $
 +
when $  x = - y \neq x _ {0} $,  
 +
and satisfies given boundary conditions on $  \sigma $,  
 +
$  \gamma _ {0} $
 +
and $  \gamma _ {1} $.  
 +
The uniqueness and existence of a solution of this problem, both for equation (5) and for more general equations, can be proved under certain geometric conditions on the boundary of $  \Omega $,  
 +
especially on the curve $  \sigma $.  
 +
The general mixed Bitsadze problem can be regarded as completely solved in the special case when $  \Gamma _ {1} $
 +
coincides $  ( x _ {0} = 1 ) $
 +
with the characteristic $  B C $
 +
through the point $  B $.  
 +
An important consequence of the fact that the general mixed Bitsadze problem is correctly posed, in the case of equation (5) for example, is that the [[Dirichlet problem|Dirichlet problem]] for mixed domains of the form $  \Omega $
 +
is incorrectly posed, whatever the size and form of the domain $  \Omega  ^ {-} $
 +
of hyperbolicity.
  
 
For a fairly large class of linear equations
 
For a fairly large class of linear equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250117.png" /></td> </tr></table>
+
$$
 +
k ( y) u _ {xx} + u _ {yy} + a u _ {x} + b u _ {y} + c u  = f
 +
$$
  
it is known that the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250118.png" /> has a substantial influence on the correctness of posing the Dirichlet problem in corresponding mixed domains of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250119.png" />.
+
it is known that the coefficient $  a ( x , y ) $
 +
has a substantial influence on the correctness of posing the Dirichlet problem in corresponding mixed domains of the form $  \Omega $.
  
Another type of mixed problem is the Frankl problem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250120.png" /> be a simply-connected domain with the following boundary: the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250121.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250122.png" /> of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250123.png" />, a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250124.png" /> with end points at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250126.png" /> and lying in the quadrant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250127.png" />, the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250128.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250129.png" /> of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250130.png" />, and the characteristic through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250132.png" /> of the equation of mixed type under consideration (e.g. equation (4)). The Frankl problem consists of finding a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250133.png" /> of the equation of mixed type in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250134.png" />, given the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250135.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250136.png" /> and the conditions
+
Another type of mixed problem is the Frankl problem. Let $  \Omega $
 +
be a simply-connected domain with the following boundary: the interval $  A  ^  \prime  A $:  
 +
$  - 1 \leq  y \leq  1 $
 +
of the line $  x = 0 $,  
 +
a smooth curve $  \sigma $
 +
with end points at $  A ( 0 , 1 ) $
 +
and $  B ( a , 0 ) $
 +
and lying in the quadrant $  x > 0 , y > 0 $,  
 +
the interval $  C B $:  
 +
$  a _ {1} \leq  x \leq  a $
 +
of the line $  y = 0 $,  
 +
and the characteristic through $  A  ^  \prime  ( 0 , - 1 ) $
 +
and $  C ( a _ {1} , 0 ) $
 +
of the equation of mixed type under consideration (e.g. equation (4)). The Frankl problem consists of finding a solution $  u ( x , y ) $
 +
of the equation of mixed type in $  \Omega $,  
 +
given the value of $  u ( x , y ) $
 +
on $  \sigma \cup C B $
 +
and the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250137.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250138.png" /></td> </tr></table>
+
\frac{\partial  u }{\partial  x }
 +
  = 0 ,\ \
 +
u ( 0 , y ) - u ( 0 , - y )  = f ( y) ,
 +
$$
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250139.png" />. This problem has been investigated chiefly for model equations of mixed type and has been completely solved for equation (5) in the case where the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250140.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250141.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250143.png" /> is the arc length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250144.png" /> measured from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250145.png" />.
+
$$
 +
- 1  \leq  y  \leq  1 ,\  x  = 0 ,
 +
$$
 +
 
 +
on  $  A  ^  \prime  A $.  
 +
This problem has been investigated chiefly for model equations of mixed type and has been completely solved for equation (5) in the case where the curve $  \sigma $:  
 +
$  x= x ( s) , y = y ( s) $
 +
is such that $  d y / d s \geq  0 $,  
 +
where $  s $
 +
is the arc length of $  \sigma $
 +
measured from the point $  B ( a , 0 ) $.
  
 
Basic boundary value problems have been formulated for equations of mixed type of the first kind and, adapted with appropriate modifications, for equations of mixed type of the second kind. These modifications are necessary because the Dirichlet problem for elliptic equations with characteristic degeneracy is not always correctly posed.
 
Basic boundary value problems have been formulated for equations of mixed type of the first kind and, adapted with appropriate modifications, for equations of mixed type of the second kind. These modifications are necessary because the Dirichlet problem for elliptic equations with characteristic degeneracy is not always correctly posed.
  
In the formulation of the boundary value problems for equation (1) in mixed domains, a new aspect is introduced if the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250146.png" /> of change of type is also a line of degeneracy of the order of equation, which occurs, for example, in the case of the equation
+
In the formulation of the boundary value problems for equation (1) in mixed domains, a new aspect is introduced if the line $  \delta $
 +
of change of type is also a line of degeneracy of the order of equation, which occurs, for example, in the case of the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250147.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
y  ^ {2p} u _ {xx} + yu _ {yy} + \beta u _ {y}  = 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250148.png" /> is a natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250149.png" /> is a constant such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250150.png" />.
+
where $  p $
 +
is a natural number and $  \beta $
 +
is a constant such that $  1 - 2 p \leq  2 \beta < 1 $.
  
For equations (5), (6), (7), there are, in addition to the above, a number of essentially new boundary value problems. These are chiefly characterized by the fact that the entire boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250151.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250152.png" /> (where the Tricomi problem is posed) carries the following boundary conditions: the Dirichlet conditions, for example, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250153.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250154.png" />, with some non-local condition pointwise connecting the values of the desired solution or a (fractional) derivative of it of a certain order. In particular, these problems include a simple example of a correctly-posed self-adjoint mixed boundary value problem.
+
For equations (5), (6), (7), there are, in addition to the above, a number of essentially new boundary value problems. These are chiefly characterized by the fact that the entire boundary $  \sigma \cup A C \cup B C $
 +
of $  \Omega $ (where the Tricomi problem is posed) carries the following boundary conditions: the Dirichlet conditions, for example, on $  \Sigma $
 +
and on $  A C \cup B C $,  
 +
with some non-local condition pointwise connecting the values of the desired solution or a (fractional) derivative of it of a certain order. In particular, these problems include a simple example of a correctly-posed self-adjoint mixed boundary value problem.
  
 
Boundary value problems have also been studied for equations (and systems) of mixed type in domains containing in their interiors several lines of degeneracy of type, or one single closed parabolic line.
 
Boundary value problems have also been studied for equations (and systems) of mixed type in domains containing in their interiors several lines of degeneracy of type, or one single closed parabolic line.
Line 87: Line 266:
 
Significant difficulties arise in the search for well-posed problems for equations of mixed type with many variables. Nevertheless, several important results have been obtained also in this direction. For the equation
 
Significant difficulties arise in the search for well-posed problems for equations of mixed type with many variables. Nevertheless, several important results have been obtained also in this direction. For the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250155.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
(  \mathop{\rm sign}  z ) \cdot u _ {xx} + u _ {yy} + u _ {zz}  = \
 +
f ( x , y , z ) ,
 +
$$
  
which is a simple model of an equation of mixed type having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250156.png" /> as a time-like plane of degeneracy of type, the following problem is known to be correctly posed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250157.png" /> be a finite simply-connected three-dimensional domain, bounded by a piecewise smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250158.png" /> and by the characteristic surfaces
+
which is a simple model of an equation of mixed type having $  z = 0 $
 +
as a time-like plane of degeneracy of type, the following problem is known to be correctly posed. Let $  \Omega $
 +
be a finite simply-connected three-dimensional domain, bounded by a piecewise smooth surface $  z = f ( x , y ) \geq  0 $
 +
and by the characteristic surfaces
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250159.png" /></td> </tr></table>
+
$$
 +
S _ {1} : x + x _ {0= \sqrt {y  ^ {2} + z  ^ {2} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250160.png" /></td> </tr></table>
+
$$
 +
S _ {2} : x - x _ {0= \sqrt {y  ^ {2} + z  ^ {2} }
 +
$$
  
of equation (8). One has to find a continuously differentiable function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250161.png" />, satisfying equation (8) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250162.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250163.png" />, that vanishes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250164.png" /> and on one of the characteristic surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250166.png" />. Existence of a weak solution and uniqueness of a strong solution for this problem have been proved for the more general equation
+
of equation (8). One has to find a continuously differentiable function in $  \Omega $,  
 +
satisfying equation (8) in $  \Omega $
 +
for $  z \neq 0 $,  
 +
that vanishes on $  \sigma $
 +
and on one of the characteristic surfaces $  S _ {1} $,  
 +
$  S _ {2} $.  
 +
Existence of a weak solution and uniqueness of a strong solution for this problem have been proved for the more general equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250167.png" /></td> </tr></table>
+
$$
 +
(  \mathop{\rm sign}  x _ {n} ) \cdot u _ {x _ {n}  x _ {n} } +
 +
\Delta _ {x} u  = f ( x _ {0} , x ) ,\  x = ( x _ {1} \dots x _ {n} ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250168.png" /> is the [[Laplace operator|Laplace operator]] in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250169.png" />.
+
where $  \Delta _ {x} $
 +
is the [[Laplace operator|Laplace operator]] in the variables $  x _ {1} \dots x _ {n} $.
  
 
For the equation
 
For the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250170.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
$$ \tag{9 }
 +
x _ {0}  ^ {2m} \Delta _ {x} u -
 +
x _ {0} u _ {x _ {0}  x _ {0} } +
 +
\left ( m -  
 +
\frac{1}{2}
 +
\right ) u _ {x _ {0}  }  =  0
 +
$$
  
with part of the space-like hyperplane of degeneracy both of type and of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250171.png" /> contained in the mixed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250172.png" />, boundary value problems of a special form have been studied. Here the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250173.png" /> lying in the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250174.png" /> carries data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250175.png" />, and the part lying in the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250176.png" /> (the characteristic conoid of equation (9)) carries certain integral averages of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250177.png" />.
+
with part of the space-like hyperplane of degeneracy both of type and of order $  x = 0 $
 +
contained in the mixed domain $  \Omega $,  
 +
boundary value problems of a special form have been studied. Here the part of $  \partial  \Omega $
 +
lying in the half-space $  x _ {0} < 0 $
 +
carries data $  u ( x _ {0} , x ) $,  
 +
and the part lying in the half-space $  x _ {0} > 0 $ (the characteristic conoid of equation (9)) carries certain integral averages of $  u ( x _ {0} , x ) $.
  
 
Other model equations of mixed type in bounded and unbounded three-dimensional domains have been studied, including the equations
 
Other model equations of mixed type in bounded and unbounded three-dimensional domains have been studied, including the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250178.png" /></td> </tr></table>
+
$$
 +
z  ^ {2m+1} u _ {xx} + u _ {yy} + u _ {zz}  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250179.png" /></td> </tr></table>
+
$$
 +
z  ^ {2m+1} ( u _ {xx} + u _ {yy} ) + u _ {zz}  = 0 .
 +
$$
  
 
There is also a uniqueness criterion of the solution of the Dirichlet problem for a large class of self-adjoint equations of mixed type in cylindrical domains.
 
There is also a uniqueness criterion of the solution of the Dirichlet problem for a large class of self-adjoint equations of mixed type in cylindrical domains.
Line 117: Line 331:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bers,  "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadse,  "Equations of mixed type" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Bitsadze,  "On the theory of equations of mixed type whose order is degenerate on the line of change of type" , ''Continuum mechanics and related problems of analysis'' , Moscow  (1972)  pp. 47–52  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Bitsadze,  A.M. Nakhusev,  "Correct formulation of problems for equations of mixed type in multidimensional domains"  ''Soviet Math. Dokl.'' , '''13''' :  4  (1972)  pp. 857–860  ''Dokl. Akad. Nauk. SSSR'' , '''205''' :  1  (1972)  pp. 9–12</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.N. Vekua,  "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.D. Karatopakliev,  "A class of equations of mixed type"  ''Diff. Equations'' , '''5''' :  1  (1969)  pp. 171–176  ''Differentsial'nye Uravn.'' , '''5''' :  1  (1969)  pp. 199–205</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.V. Keldysh,  "On certain cases of degeneracy on the boundary of a domain for equations of elliptic type"  ''Dokl. Akad. Nauk SSSR'' , '''77'''  (1951)  pp. 181–183  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.S. Salakhitdinov,  "Certain boundary value problems for equations of mixed type"  ''Izv. Akad. Nauk UzbSSR, Ser. Fiz.-Mat. Nauk'' , '''1'''  (1969)  pp. 27–33  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.M. Smirnov,  "Equations of mixed type" , Amer. Math. Soc.  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.P. Soldatov,  "A problem in function theory"  ''Diff. Equations'' , '''9''' :  2  (1973)  pp. 248–253  ''Differentsial'nye Uravn.'' , '''9''' :  2  (1973)  pp. 325–332</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  F. Tricomi,  ''Atti Accad. Naz. Lincei, Ser. 5'' , '''14'''  (1932)  pp. 134–247</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  F.I. Frankl,  "Selected work on gas dynamics" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  K.O. Friedrichs,  "Symmetric positive linear differential equations"  ''Comm. Pure Appl. Math.'' , '''11'''  (1958)  pp. 333–418</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  S. Gellerstedt,  "Quelques problèmes mixtes pour l'équation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250180.png" />"  ''Ark. Mat. Astr. Fysik'' , '''26A''' :  3  (1937)  pp. 1–32</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  P. Germain,  R. Bader,  "Sur le problème de Tricomi"  ''C.R. Acad. Sci. Paris'' , '''232'''  (1951)  pp. 463–465</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bers,  "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadse,  "Equations of mixed type" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Bitsadze,  "On the theory of equations of mixed type whose order is degenerate on the line of change of type" , ''Continuum mechanics and related problems of analysis'' , Moscow  (1972)  pp. 47–52  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Bitsadze,  A.M. Nakhusev,  "Correct formulation of problems for equations of mixed type in multidimensional domains"  ''Soviet Math. Dokl.'' , '''13''' :  4  (1972)  pp. 857–860  ''Dokl. Akad. Nauk. SSSR'' , '''205''' :  1  (1972)  pp. 9–12</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.N. Vekua,  "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.D. Karatopakliev,  "A class of equations of mixed type"  ''Diff. Equations'' , '''5''' :  1  (1969)  pp. 171–176  ''Differentsial'nye Uravn.'' , '''5''' :  1  (1969)  pp. 199–205</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.V. Keldysh,  "On certain cases of degeneracy on the boundary of a domain for equations of elliptic type"  ''Dokl. Akad. Nauk SSSR'' , '''77'''  (1951)  pp. 181–183  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.S. Salakhitdinov,  "Certain boundary value problems for equations of mixed type"  ''Izv. Akad. Nauk UzbSSR, Ser. Fiz.-Mat. Nauk'' , '''1'''  (1969)  pp. 27–33  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.M. Smirnov,  "Equations of mixed type" , Amer. Math. Soc.  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.P. Soldatov,  "A problem in function theory"  ''Diff. Equations'' , '''9''' :  2  (1973)  pp. 248–253  ''Differentsial'nye Uravn.'' , '''9''' :  2  (1973)  pp. 325–332</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  F. Tricomi,  ''Atti Accad. Naz. Lincei, Ser. 5'' , '''14'''  (1932)  pp. 134–247</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  F.I. Frankl,  "Selected work on gas dynamics" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  K.O. Friedrichs,  "Symmetric positive linear differential equations"  ''Comm. Pure Appl. Math.'' , '''11'''  (1958)  pp. 333–418</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  S. Gellerstedt,  "Quelques problèmes mixtes pour l'équation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064250/m064250180.png" />"  ''Ark. Mat. Astr. Fysik'' , '''26A''' :  3  (1937)  pp. 1–32</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  P. Germain,  R. Bader,  "Sur le problème de Tricomi"  ''C.R. Acad. Sci. Paris'' , '''232'''  (1951)  pp. 463–465</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 06:19, 24 February 2022


A partial differential equation which is of varying type (elliptic, hyperbolic or parabolic) in its domain of definition. A linear (or quasi-linear) differential equation of the second order with two unknown variables,

$$ \tag{1 } A u _ {xx} + 2 B u _ {xy} + C u _ {yy} = \ f ( x , y , u , u _ {x} , u _ {y} ) $$

and with coefficients defined in the domain $ \Omega $ is an equation of mixed type if the discriminant $ \Delta = A C - B ^ {2} $ of the characteristic form

$$ A d y ^ {2} + 2 B d x d y + C d x ^ {2} = Q $$

takes the value zero in $ \Omega $ but is not identically zero there.

The curve $ \delta $ defined by the equation $ \Delta = 0 $ is called the parabolic line of equation (1) or the line of degeneracy (change) of type.

If $ \Delta $ does not change sign when the point $ ( x , y ) $ crosses the parabolic line $ \delta $ in $ \Omega $, then equation (1) is a degenerate equation of elliptic-parabolic $ ( \Delta \geq 0 ) $ or hyperbolic-parabolic $ ( \Delta \leq 0 ) $ type (see Degenerate partial differential equation).

Under certain smoothness conditions on $ A $, $ B $, $ C $, and $ \delta $, there is a non-singular real transformation of the independent variables sending equation (1) (in the case where the sign of $ \Delta $ alternates in the neighbourhood of a chosen point of $ \delta $ where $ A ^ {2} + B ^ {2} \neq 0 $) to one of the following canonical forms (the notation for the independent variables is preserved):

$$ \tag{2 } y ^ {2m+1} u _ {xx} + u _ {yy} = \ F ( x , y , u , u _ {x} , u _ {y} ) , $$

$$ \tag{3 } u _ {xx} + y ^ {2m+1} u _ {yy} = F ( x , y , u , u _ {x} , u _ {y} ) . $$

The equations (2) and (3) are equations of mixed (elliptic-hyperbolic) type in any domain containing a segment of the line of degeneracy $ y = 0 $.

The domain $ \Omega $ of definition of an equation of mixed type is sometimes called a mixed domain, and boundary value problems in mixed domains are called mixed boundary value problems. The part $ \Omega ^ {+} $ ($ \Omega ^ {-} $) of a mixed domain $ \Omega $ where the equation is of elliptic (hyperbolic) type is called the domain of ellipticity (hyperbolicity).

Many problems of an applied nature reduce to finding specific solutions of equations of mixed type; in particular, problems of plane transonic flow of a compressible medium, and problems in the theory of envelopes.

An equation (1) of mixed type is called an equation of the first kind (second kind) if the characteristic form $ Q \neq 0 $ ($ Q = 0 $) everywhere on the parabolic line. The Chaplygin equation

$$ \tag{4 } k ( y) u _ {xx} + u _ {yy} = 0 , $$

where $ k ( y) $ is a continuously-differentiable monotone function such that $ y k ( y) > 0 $ when $ y \neq 0 $, is a typical example of an equation of mixed type of the first kind. When $ k ( y) = y $, equation (4) is usually called the Tricomi equation.

An important model for an equation of mixed type (with a discontinuous coefficient in front of one of the higher derivatives) is the equation of Lavrent'ev–Bitsadze

$$ \tag{5 } ( \mathop{\rm sign} y ) \cdot u _ {xx} + u _ {yy} = 0 . $$

One of the basic boundary value problems for equations of mixed type (of the first kind) is the Tricomi problem, which is as follows for an equation of the form (2). Let $ \Omega $ be a finite simply-connected domain in the Euclidean plane with independent variables $ x $ and $ y $, bounded by a simple Jordan curve $ \sigma $ with end points $ A ( 0 , 0 ) $, $ B ( 1 , 0 ) $ lying in the half-plane $ y > 0 $, and by the parts $ A C $ and $ B C $ of the characteristics of equation (2) going through the point $ C ( 1/2 , y _ {C} ) $, $ y _ {C} < 0 $. The Tricomi problem is to find a solution $ u ( x , y ) $ of equation (2) that is continuous in the closure $ \overline \Omega \; $ of $ \Omega $ and takes given values on the curve $ \sigma \cup A C $.

In the theory of the Tricomi problem, an essential role is played by the Bitsadze extremum principle, which, in the case of equation (5), states that a solution $ u ( x , y ) $ of equation (5) in the class $ C ( \overline \Omega ) \cap C ^ {1} ( \Omega ) $ which vanishes on the characteristic $ A C $: $ x + y = 0 $, $ 0 \leq x \leq 1/2 $, in the closure $ \overline \Omega ^ {+} $ of the domain of ellipticity $ \Omega ^ {+} = \Omega \cap \{ y > 0 \} $, attains its extremum on the curve $ \sigma $.

This principle, which guarantees the uniqueness and stability of a solution of the Tricomi problem (and also provides a basic estimate to prove existence by the alternating method), can be generalized to a very wide class of linear and quasi-linear equations of mixed type. In particular, it applies to Chaplygin equations (and Tricomi equations) whenever $ k ( y) $ is twice continuously differentiable and $ 5 k ^ {\prime 2 } \geq 4 k k ^ \prime $ for $ y < 0 $. The Bitsadze extremum principle also holds for the equation

$$ \tag{6 } \mathop{\rm sign} y \cdot | y | ^ \alpha u _ {xx} + u _ {yy} = 0 , \alpha = \textrm{ const } > 0 . $$

The solution of the Tricomi problem for equation (6) in the corresponding mixed domain $ \Omega $ can be written in explicit form when the elliptic part $ \sigma $ of the boundary of this domain coincides with the so-called normal contour $ \sigma _ {0} $:

$$ x ^ {2} + \left ( \frac{2}{\alpha + 2 } \right ) ^ {2} y ^ {\alpha + 2 } = \frac{1}{4} . $$

In the general case, under specific conditions on the curve $ \sigma $ and on the class in which the solutions are sought, the Tricomi problem for equation (6) reduces to an equivalent singular integral equation which is (by virtue of the uniqueness condition) unconditionally solvable. The method of integral equations can also be applied to prove existence of a solution of the Tricomi problem, and of other mixed problems, for more general equations of the form

$$ \mathop{\rm sign} y \cdot | y | ^ \alpha u _ {xx} + u _ {yy} = F ( x , y , u , u _ {x} , u _ {y} ) $$

with power degeneracy of order $ \alpha $.

Function-theoretical methods and functional analysis, in particular the use of a priori estimates, have made it possible to extend significantly the class of equations of mixed type and mixed domains for which existence and uniqueness of a (generalized) solution can be proved, both for the Tricomi problems and for various other mixed problems.

An important generalization of the Tricomi problem is the general mixed Bitsadze problem, which, in the case of equation (5), can be posed as follows. Let $ \Omega $ be a simply-connected mixed domain bounded by a simple Jordan curve $ \sigma $ lying in the half-plane $ y > 0 $ with end points $ A ( 0 , 0 ) $, $ B ( 1 , 0 ) $ and (smooth) monotone curves $ \Gamma _ {0} $ and $ \Gamma _ {1} $ through these points, meeting at a point $ C ( x _ {1} , y _ {1} ) $, $ y _ {1} < 0 $. It is assumed that $ \Gamma _ {0} $ and $ \Gamma _ {1} $ lie in the domain bounded by the characteristics $ x + y = 0 $, $ x - y = 1 $ and the interval $ A B $: $ 0 \leq x \leq 1 $ on the $ x $-axis. Let $ B _ {0} $ and $ B _ {1} $ denote the points of intersection of the characteristics $ x - y = x _ {0} $ and $ x + y = x _ {0} $ with the curves $ \Gamma _ {0} $ and $ \Gamma _ {1} $, where $ x _ {0} $ is any fixed point of the semi-interval $ x _ {1} + y _ {1} < x _ {1} \leq x _ {1} - y _ {1} $, and let $ \gamma _ {0} $ and $ \gamma _ {1} $ denote the parts of $ \Gamma _ {0} $ and $ \Gamma _ {1} $ lying between $ A , B _ {0} $, and $ B , B _ {1} $, respectively. The general mixed Bitsadze problem consists of finding a regular solution (when $ y \neq 0 $, $ x \pm y \neq x _ {0} $) of equation (5) in $ \Omega $ which is continuous in $ \overline \Omega \; $, has continuous first derivatives in $ \Omega $ when $ x = - y \neq x _ {0} $, and satisfies given boundary conditions on $ \sigma $, $ \gamma _ {0} $ and $ \gamma _ {1} $. The uniqueness and existence of a solution of this problem, both for equation (5) and for more general equations, can be proved under certain geometric conditions on the boundary of $ \Omega $, especially on the curve $ \sigma $. The general mixed Bitsadze problem can be regarded as completely solved in the special case when $ \Gamma _ {1} $ coincides $ ( x _ {0} = 1 ) $ with the characteristic $ B C $ through the point $ B $. An important consequence of the fact that the general mixed Bitsadze problem is correctly posed, in the case of equation (5) for example, is that the Dirichlet problem for mixed domains of the form $ \Omega $ is incorrectly posed, whatever the size and form of the domain $ \Omega ^ {-} $ of hyperbolicity.

For a fairly large class of linear equations

$$ k ( y) u _ {xx} + u _ {yy} + a u _ {x} + b u _ {y} + c u = f $$

it is known that the coefficient $ a ( x , y ) $ has a substantial influence on the correctness of posing the Dirichlet problem in corresponding mixed domains of the form $ \Omega $.

Another type of mixed problem is the Frankl problem. Let $ \Omega $ be a simply-connected domain with the following boundary: the interval $ A ^ \prime A $: $ - 1 \leq y \leq 1 $ of the line $ x = 0 $, a smooth curve $ \sigma $ with end points at $ A ( 0 , 1 ) $ and $ B ( a , 0 ) $ and lying in the quadrant $ x > 0 , y > 0 $, the interval $ C B $: $ a _ {1} \leq x \leq a $ of the line $ y = 0 $, and the characteristic through $ A ^ \prime ( 0 , - 1 ) $ and $ C ( a _ {1} , 0 ) $ of the equation of mixed type under consideration (e.g. equation (4)). The Frankl problem consists of finding a solution $ u ( x , y ) $ of the equation of mixed type in $ \Omega $, given the value of $ u ( x , y ) $ on $ \sigma \cup C B $ and the conditions

$$ \frac{\partial u }{\partial x } = 0 ,\ \ u ( 0 , y ) - u ( 0 , - y ) = f ( y) , $$

$$ - 1 \leq y \leq 1 ,\ x = 0 , $$

on $ A ^ \prime A $. This problem has been investigated chiefly for model equations of mixed type and has been completely solved for equation (5) in the case where the curve $ \sigma $: $ x= x ( s) , y = y ( s) $ is such that $ d y / d s \geq 0 $, where $ s $ is the arc length of $ \sigma $ measured from the point $ B ( a , 0 ) $.

Basic boundary value problems have been formulated for equations of mixed type of the first kind and, adapted with appropriate modifications, for equations of mixed type of the second kind. These modifications are necessary because the Dirichlet problem for elliptic equations with characteristic degeneracy is not always correctly posed.

In the formulation of the boundary value problems for equation (1) in mixed domains, a new aspect is introduced if the line $ \delta $ of change of type is also a line of degeneracy of the order of equation, which occurs, for example, in the case of the equation

$$ \tag{7 } y ^ {2p} u _ {xx} + yu _ {yy} + \beta u _ {y} = 0 , $$

where $ p $ is a natural number and $ \beta $ is a constant such that $ 1 - 2 p \leq 2 \beta < 1 $.

For equations (5), (6), (7), there are, in addition to the above, a number of essentially new boundary value problems. These are chiefly characterized by the fact that the entire boundary $ \sigma \cup A C \cup B C $ of $ \Omega $ (where the Tricomi problem is posed) carries the following boundary conditions: the Dirichlet conditions, for example, on $ \Sigma $ and on $ A C \cup B C $, with some non-local condition pointwise connecting the values of the desired solution or a (fractional) derivative of it of a certain order. In particular, these problems include a simple example of a correctly-posed self-adjoint mixed boundary value problem.

Boundary value problems have also been studied for equations (and systems) of mixed type in domains containing in their interiors several lines of degeneracy of type, or one single closed parabolic line.

Analogues of the Tricomi problem have been studied for certain classes of equations and systems of mixed type in two independent variables, and for equations of higher order.

Significant difficulties arise in the search for well-posed problems for equations of mixed type with many variables. Nevertheless, several important results have been obtained also in this direction. For the equation

$$ \tag{8 } ( \mathop{\rm sign} z ) \cdot u _ {xx} + u _ {yy} + u _ {zz} = \ f ( x , y , z ) , $$

which is a simple model of an equation of mixed type having $ z = 0 $ as a time-like plane of degeneracy of type, the following problem is known to be correctly posed. Let $ \Omega $ be a finite simply-connected three-dimensional domain, bounded by a piecewise smooth surface $ z = f ( x , y ) \geq 0 $ and by the characteristic surfaces

$$ S _ {1} : x + x _ {0} = \sqrt {y ^ {2} + z ^ {2} } , $$

$$ S _ {2} : x - x _ {0} = \sqrt {y ^ {2} + z ^ {2} } $$

of equation (8). One has to find a continuously differentiable function in $ \Omega $, satisfying equation (8) in $ \Omega $ for $ z \neq 0 $, that vanishes on $ \sigma $ and on one of the characteristic surfaces $ S _ {1} $, $ S _ {2} $. Existence of a weak solution and uniqueness of a strong solution for this problem have been proved for the more general equation

$$ ( \mathop{\rm sign} x _ {n} ) \cdot u _ {x _ {n} x _ {n} } + \Delta _ {x} u = f ( x _ {0} , x ) ,\ x = ( x _ {1} \dots x _ {n} ) , $$

where $ \Delta _ {x} $ is the Laplace operator in the variables $ x _ {1} \dots x _ {n} $.

For the equation

$$ \tag{9 } x _ {0} ^ {2m} \Delta _ {x} u - x _ {0} u _ {x _ {0} x _ {0} } + \left ( m - \frac{1}{2} \right ) u _ {x _ {0} } = 0 $$

with part of the space-like hyperplane of degeneracy both of type and of order $ x = 0 $ contained in the mixed domain $ \Omega $, boundary value problems of a special form have been studied. Here the part of $ \partial \Omega $ lying in the half-space $ x _ {0} < 0 $ carries data $ u ( x _ {0} , x ) $, and the part lying in the half-space $ x _ {0} > 0 $ (the characteristic conoid of equation (9)) carries certain integral averages of $ u ( x _ {0} , x ) $.

Other model equations of mixed type in bounded and unbounded three-dimensional domains have been studied, including the equations

$$ z ^ {2m+1} u _ {xx} + u _ {yy} + u _ {zz} = 0 , $$

$$ z ^ {2m+1} ( u _ {xx} + u _ {yy} ) + u _ {zz} = 0 . $$

There is also a uniqueness criterion of the solution of the Dirichlet problem for a large class of self-adjoint equations of mixed type in cylindrical domains.

References

[1] L. Bers, "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley (1958)
[2] A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)
[3] A.V. Bitsadze, "On the theory of equations of mixed type whose order is degenerate on the line of change of type" , Continuum mechanics and related problems of analysis , Moscow (1972) pp. 47–52 (In Russian)
[4] A.V. Bitsadze, A.M. Nakhusev, "Correct formulation of problems for equations of mixed type in multidimensional domains" Soviet Math. Dokl. , 13 : 4 (1972) pp. 857–860 Dokl. Akad. Nauk. SSSR , 205 : 1 (1972) pp. 9–12
[5] I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian)
[6] G.D. Karatopakliev, "A class of equations of mixed type" Diff. Equations , 5 : 1 (1969) pp. 171–176 Differentsial'nye Uravn. , 5 : 1 (1969) pp. 199–205
[7] M.V. Keldysh, "On certain cases of degeneracy on the boundary of a domain for equations of elliptic type" Dokl. Akad. Nauk SSSR , 77 (1951) pp. 181–183 (In Russian)
[8] M.S. Salakhitdinov, "Certain boundary value problems for equations of mixed type" Izv. Akad. Nauk UzbSSR, Ser. Fiz.-Mat. Nauk , 1 (1969) pp. 27–33 (In Russian)
[9] M.M. Smirnov, "Equations of mixed type" , Amer. Math. Soc. (1978) (Translated from Russian)
[10] A.P. Soldatov, "A problem in function theory" Diff. Equations , 9 : 2 (1973) pp. 248–253 Differentsial'nye Uravn. , 9 : 2 (1973) pp. 325–332
[11] F. Tricomi, Atti Accad. Naz. Lincei, Ser. 5 , 14 (1932) pp. 134–247
[12] F.I. Frankl, "Selected work on gas dynamics" , Moscow (1973) (In Russian)
[13] K.O. Friedrichs, "Symmetric positive linear differential equations" Comm. Pure Appl. Math. , 11 (1958) pp. 333–418
[14] S. Gellerstedt, "Quelques problèmes mixtes pour l'équation " Ark. Mat. Astr. Fysik , 26A : 3 (1937) pp. 1–32
[15] P. Germain, R. Bader, "Sur le problème de Tricomi" C.R. Acad. Sci. Paris , 232 (1951) pp. 463–465

Comments

More recent workers tend to favour functional-analytic methods, see [a1]. For a constructive approach, using Fourier integral operator methods, see [a2].

References

[a1] M. Schneider, "Ueber Differentialgleichungen zweiter Ordnung vom gemischten Typ im " Math. Nachr. , 66 (1975) pp. 57–66
[a2] R.J.P. Groothhuizen, "Mixed elliptic-hyperbolic partial differential operators: a case-study in Fourier integral operators" , CWI Tracts , 16 , CWI , Amsterdam (1985) (Thesis Free University Amsterdam)
How to Cite This Entry:
Mixed-type differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed-type_differential_equation&oldid=14168
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article