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The problem of finding a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173501.png" />, regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173502.png" />, to an elliptic equation
+
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<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/b/b017/b017350/b0173503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
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which satisfies certain additional conditions on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173505.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b0173509.png" /> are given functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735010.png" />.
+
The problem of finding a solution  $  u $,  
 +
regular in a domain  $  D $,  
 +
to an elliptic equation
  
The classical boundary value problems are special cases of the following problem: Find a solution to equation (1), regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735011.png" /> and satisfying on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735012.png" />
+
$$ \tag{1 }
 +
\sum _ {i, k = 0 } ^ { n }
 +
a _ {ik}
 +
\frac{\partial  ^ {2} u }{\partial  x _ {i} \partial  x _ {k} }
 +
+
 +
\sum _ {i = 0 } ^ { n }
 +
b _ {i}
 +
\frac{\partial  u }{\partial  x _ {i} }
 +
+
 +
cu  = f,
 +
$$
  
<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/b/b017/b017350/b01735013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
which satisfies certain additional conditions on the boundary  $  \Gamma $
 +
of  $  D $.  
 +
Here  $  a _ {ik} $,
 +
b _ {i} $,
 +
$  c $
 +
and  $  f $
 +
are given functions on  $  D $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735014.png" /> denotes differentiation in some direction, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735016.png" /> are given continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735017.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735018.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735019.png" /> (see [[#References|[1]]]).
+
The classical boundary value problems are special cases of the following problem: Find a solution to equation (1), regular in a domain  $  D $
 +
and satisfying on  $  \Gamma $
  
Putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735020.png" />, one obtains the [[Dirichlet problem|Dirichlet problem]]; with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735021.png" /> one has a problem with oblique derivative (see [[Differential equation, partial, oblique derivatives|Differential equation, partial, oblique derivatives]]), which becomes a [[Neumann problem]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735022.png" /> is the direction of the conormal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735025.png" /> are disjoint open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735027.png" /> is either empty or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735028.png" />-dimensional manifold, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735034.png" />, one obtains a [[Mixed problem|mixed problem]].
+
$$ \tag{2 }
 +
a  
 +
\frac{du }{dl }
 +
+
 +
bu  = g
 +
$$
  
Problem (2) has been studied for elliptic equations in two independent variables (see [[#References|[2]]]). Fairly complete investigations have been made of the Dirichlet problem for elliptic equations in any finite number of independent variables (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]) and the problem with oblique derivative in case the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735035.png" /> is not contained in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735036.png" /> at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735037.png" />. In that case the problem with oblique derivative is a Fredholm problem and the solution is smooth to the same order as the field of directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735038.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735039.png" /> (see [[#References|[1]]]). The case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735040.png" /> lies in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735041.png" /> at certain points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735042.png" /> has been studied (see [[#References|[3]]]). The local properties of solutions to the problem with oblique derivative have been investigated (see [[#References|[5]]]). At points where the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735043.png" /> lies in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735044.png" />, the solution of the problem is less smooth than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735046.png" />. This has been used as a basis for investigating the problem in a generalized setting (see [[#References|[7]]], [[#References|[8]]]).
+
where  $  d/dl $
 +
denotes differentiation in some direction, and a, b $
 +
and  $  g $
 +
are given continuous functions on  $  \Gamma $
 +
with | a | + | b | > 0 $
 +
everywhere on  $  \Gamma $(
 +
see [[#References|[1]]]).
  
Consider the following boundary problem for harmonic functions regular in the unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735047.png" />:
+
Putting  $  a = 0, b = 1 $,
 +
one obtains the [[Dirichlet problem|Dirichlet problem]]; with  $  b = 0, a = 1 $
 +
one has a problem with oblique derivative (see [[Differential equation, partial, oblique derivatives|Differential equation, partial, oblique derivatives]]), which becomes a [[Neumann problem]] if  $  l $
 +
is the direction of the conormal. If  $  \Gamma = \overline \Gamma \; _ {1} \cup \overline \Gamma \; _ {2} $,
 +
where  $  \Gamma _ {1} $
 +
and  $  \Gamma _ {2} $
 +
are disjoint open subsets of  $  \Gamma $,
 +
and  $  \overline \Gamma \; _ {1} \cap \overline \Gamma \; _ {2} $
 +
is either empty or an  $  (n - 2) $-
 +
dimensional manifold, with  $  a = 1 $,
 +
$  b = 0 $
 +
on  $  \Gamma _ {1} $,
 +
$  a = 0 $,
 +
b = 1 $
 +
on  $  \Gamma _ {2} $,
 +
one obtains a [[Mixed problem|mixed problem]].
  
<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/b/b017/b017350/b01735048.png" /></td> </tr></table>
+
Problem (2) has been studied for elliptic equations in two independent variables (see [[#References|[2]]]). Fairly complete investigations have been made of the Dirichlet problem for elliptic equations in any finite number of independent variables (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]) and the problem with oblique derivative in case the direction  $  l $
 +
is not contained in a tangent plane to  $  \Gamma $
 +
at any point of  $  \Gamma $.  
 +
In that case the problem with oblique derivative is a Fredholm problem and the solution is smooth to the same order as the field of directions  $  l $
 +
and the function  $  g $(
 +
see [[#References|[1]]]). The case in which  $  l $
 +
lies in a tangent plane to  $  \Gamma $
 +
at certain points of  $  \Gamma $
 +
has been studied (see [[#References|[3]]]). The local properties of solutions to the problem with oblique derivative have been investigated (see [[#References|[5]]]). At points where the field  $  l $
 +
lies in a tangent plane to  $  \Gamma $,
 +
the solution of the problem is less smooth than  $  l $
 +
and  $  g $.
 +
This has been used as a basis for investigating the problem in a generalized setting (see [[#References|[7]]], [[#References|[8]]]).
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735049.png" /> be the set of points of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735050.png" /> at which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735051.png" /> vanishes. The vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735052.png" /> lies in a tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735053.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735054.png" />. Suppose in addition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735055.png" /> is the union of a finite number of disjoint curves; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735056.png" /> be the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735057.png" /> consisting of those points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735058.png" /> makes an acute angle with the projection of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735060.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735061.png" /> be the remaining part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735062.png" />. A generalized formulation of the problem is obtained when the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735063.png" /> are also prescribed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735064.png" />, whereas on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735065.png" /> the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735066.png" /> is allowed to have integrable singularities. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735067.png" /> is empty, the solution to the generalized problem may be made arbitrarily smooth by increasing the smoothness of the additional data of the problem. Generally speaking, a solution to the mixed problem on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735068.png" /> has singularities (see [[#References|[1]]]). In order to eliminate such singularities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735069.png" />, one must impose additional conditions on the data (see [[#References|[11]]]).
+
Consider the following boundary problem for harmonic functions regular in the unit ball  $  \Sigma \subset \mathbf R  ^ {3} $:
  
A large category of boundary value problems is constituted by what are known as problems with free boundaries. In these problems one must find not only a solution of equation (1), but also the domain in which it is regular. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735070.png" /> of the domain is unknown, but two boundary conditions must be satisfied on it. An example of this type of problem is the problem of wave motions of an ideal fluid: Find a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735071.png" />, regular in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735072.png" />, where part of the boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735073.png" /> say, is known and the normal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735074.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735075.png" />; the other part of the boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735076.png" />, is unknown and on it one gives two boundary conditions:
+
$$
 +
au _ {x} +
 +
bu _ {y} +
 +
cu _ {z}  = g;
 +
$$
  
<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/b/b017/b017350/b01735077.png" /></td> </tr></table>
+
let  $  K $
 +
be the set of points of the unit sphere  $  S $
 +
at which the function  $  \omega = ax + by + cz $
 +
vanishes. The vector field  $  P (a, b, c) $
 +
lies in a tangent plane to  $  S $
 +
at the points of  $  K $.
 +
Suppose in addition that  $  K $
 +
is the union of a finite number of disjoint curves; let  $  K  ^ {+} $
 +
be the subset of  $  K $
 +
consisting of those points at which  $  \mathop{\rm grad}  \omega $
 +
makes an acute angle with the projection of the field  $  P $
 +
on  $  S $,
 +
and let  $  K  ^ {-} $
 +
be the remaining part of  $  K $.
 +
A generalized formulation of the problem is obtained when the values of  $  u $
 +
are also prescribed on  $  K  ^ {+} $,
 +
whereas on  $  K  ^ {-} $
 +
the solution  $  u $
 +
is allowed to have integrable singularities. If  $  K  ^ {-} $
 +
is empty, the solution to the generalized problem may be made arbitrarily smooth by increasing the smoothness of the additional data of the problem. Generally speaking, a solution to the mixed problem on the set  $  \Gamma _ {0} = \overline \Gamma \; _ {1} \cap \overline \Gamma \; _ {2} $
 +
has singularities (see [[#References|[1]]]). In order to eliminate such singularities on  $  \Gamma _ {0} $,
 +
one must impose additional conditions on the data (see [[#References|[11]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735078.png" /> is a given function.
+
A large category of boundary value problems is constituted by what are known as problems with free boundaries. In these problems one must find not only a solution of equation (1), but also the domain in which it is regular. The boundary  $  \Gamma $
 +
of the domain is unknown, but two boundary conditions must be satisfied on it. An example of this type of problem is the problem of wave motions of an ideal fluid: Find a harmonic function  $  u $,
 +
regular in some domain  $  D $,
 +
where part of the boundary,  $  \Gamma _ {1} $
 +
say, is known and the normal derivative  $  \partial  u/ \partial  n $
 +
is given on  $  \Gamma _ {1} $;
 +
the other part of the boundary,  $  \Gamma _ {2} $,
 +
is unknown and on it one gives two boundary conditions:
 +
 
 +
$$
 +
 
 +
\frac{\partial  u }{\partial  n }
 +
  = 0,\ \
 +
u _ {x}  ^ {2} + u _ {y}  ^ {2} +
 +
u _ {z}  ^ {2}  = q (x, y, z),
 +
$$
 +
 
 +
where  $  q > 0 $
 +
is a given function.
  
 
For harmonic functions of two independent variables, one uses conformal mapping (see [[#References|[12]]], [[#References|[13]]], [[#References|[14]]]). See also [[Differential equation, partial, free boundaries|Differential equation, partial, free boundaries]].
 
For harmonic functions of two independent variables, one uses conformal mapping (see [[#References|[12]]], [[#References|[13]]], [[#References|[14]]]). See also [[Differential equation, partial, free boundaries|Differential equation, partial, free boundaries]].
  
The following problem has been investigated: Find a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735079.png" />, regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735080.png" /> and satisfying the condition
+
The following problem has been investigated: Find a harmonic function $  u $,  
 +
regular in a domain $  D $
 +
and satisfying the condition
  
<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/b/b017/b017350/b01735081.png" /></td> </tr></table>
+
$$
 +
|  \mathop{\rm grad}  u |  ^ {2}  = q,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735082.png" /> is a given function, on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735083.png" />. There is a complete solution of this problem for harmonic functions of two independent variables (see [[#References|[14]]]).
+
where $  q > 0 $
 +
is a given function, on the boundary $  \Gamma $.  
 +
There is a complete solution of this problem for harmonic functions of two independent variables (see [[#References|[14]]]).
  
Given an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735085.png" /> is an operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735086.png" />, uniformly elliptic in the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735087.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735088.png" />, consider the problem of determining a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735089.png" />, regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735090.png" /> and satisfying on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735092.png" /> the conditions
+
Given an equation $  Lu = f $,  
 +
where $  L $
 +
is an operator of order $  2m $,  
 +
uniformly elliptic in the closure $  \overline{D}\; $
 +
of a domain $  D $,  
 +
consider the problem of determining a solution $  u $,  
 +
regular in $  D $
 +
and satisfying on the boundary $  \Gamma $
 +
of $  D $
 +
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/b/b017/b017350/b01735093.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
B _ {j} u  = \Phi _ {j} ,\ \
 +
j = 1 \dots m,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735094.png" />, are differential operators satisfying the following complementarity condition.
+
where $  B _ {j} (x, D), j = 1 \dots m $,  
 +
are differential operators satisfying the following complementarity condition.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735095.png" /> be the principal part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735096.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735097.png" /> be the principal part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b01735099.png" /> the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350100.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350102.png" /> an arbitrary vector parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350103.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350104.png" /> denote the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350105.png" /> with positive imaginary parts. The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350107.png" />, as functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350108.png" />, must be linearly independent modulo the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350109.png" />. In this case, too, the problem is normally solvable. Violation of the complementarity condition may entail an essential change in the nature of the problem (see [[#References|[17]]]).
+
Let $  L  ^  \prime  (x, \partial  / \partial  x _ {1} \dots \partial  / \partial  x _ {n + 1 }  ) $
 +
be the principal part of $  L $,  
 +
let $  B _ {j} ^ { \prime } $
 +
be the principal part of $  B _ {j} $,  
 +
$  n $
 +
the normal to $  \Gamma $
 +
at a point $  x $
 +
and $  \lambda \neq 0 $
 +
an arbitrary vector parallel to $  \Gamma $.  
 +
Let $  \tau _ {k}  ^ {+} ( \lambda ) $
 +
denote the roots of $  L  ^  \prime  (x, \lambda + \tau n) $
 +
with positive imaginary parts. The polynomials $  B _ {j} ^ { \prime } (x, \lambda + \tau n) $,  
 +
$  j = 1 \dots m $,  
 +
as functions of $  \tau $,  
 +
must be linearly independent modulo the polynomial $  \prod _ {k = 1 }  ^ {m} ( \tau - \tau _ {k}  ^ {+} ( \lambda )) $.  
 +
In this case, too, the problem is normally solvable. Violation of the complementarity condition may entail an essential change in the nature of the problem (see [[#References|[17]]]).
  
Problem (2) is a special case of problem (3). For problem (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350110.png" />, the complementarity condition is equivalent to the condition that there be no point on the boundary of the domain at which the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350111.png" /> lies in a tangent plane to the boundary.
+
Problem (2) is a special case of problem (3). For problem (2) with $  a \equiv 1 $,  
 +
the complementarity condition is equivalent to the condition that there be no point on the boundary of the domain at which the direction $  l $
 +
lies in a tangent plane to the boundary.
  
 
Another particular case of problem (3) is the boundary value problem
 
Another particular case of problem (3) is the boundary value problem
  
<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/b/b017/b017350/b017350112.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  ^ {j} u }{\partial  n  ^ {j} }
 +
 
 +
= \Phi _ {j} ,\ \
 +
j = 0 \dots m - 1,
 +
$$
  
 
which is an analogue, to some extent, of the Dirichlet problem for higher-order elliptic equations.
 
which is an analogue, to some extent, of the Dirichlet problem for higher-order elliptic equations.
  
The boundary value problem has been studied for the poly-harmonic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017350/b017350113.png" /> when the boundary of the domain consists of manifolds of different dimensions (see [[#References|[15]]]).
+
The boundary value problem has been studied for the poly-harmonic equation $  \Delta  ^ {k} u = 0 $
 +
when the boundary of the domain consists of manifolds of different dimensions (see [[#References|[15]]]).
  
 
In investigations of boundary value problems for non-linear equations (e.g. the Dirichlet and Neumann problems), much importance attaches to a priori estimates, various fixed-point principles (see [[#References|[17]]], [[#References|[18]]]) and the generalization of Morse theory to the infinite-dimensional case (see [[#References|[19]]]).
 
In investigations of boundary value problems for non-linear equations (e.g. the Dirichlet and Neumann problems), much importance attaches to a priori estimates, various fixed-point principles (see [[#References|[17]]], [[#References|[18]]]) and the generalization of Morse theory to the infinite-dimensional case (see [[#References|[19]]]).
Line 57: Line 208:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) {{MR|0284700}} {{ZBL|0198.14101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) {{MR|0152665}} {{MR|0150320}} {{MR|0138774}} {{ZBL|0127.03505}} {{ZBL|0100.07603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) {{MR|0226183}} {{ZBL|0167.09401}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" ''Uspekhi Mat. Nauk'' : 8 (1941) pp. 171–231 (In Russian) {{MR|}} {{ZBL|0179.43901}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators and non-elliptic boundary value problems" ''Ann. of Math.'' , '''83''' : 1 (1966) pp. 129–209 {{MR|233064}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.L. Borrelli, "The singular, second order oblique derivative problem" ''J. Math. and Mech.'' , '''16''' : 1 (1966) pp. 51–81 {{MR|0203217}} {{ZBL|0143.14603}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Yu.V. Egorov, V.A. Kondrat'ev, "The oblique derivative problem" ''Mat. Sb.'' , '''78''' : 1 (1969) pp. 148–176 (In Russian) {{MR|0237953}} {{ZBL|0186.43202}} {{ZBL|0165.12202}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.G. Maz'ya, "The degenerate problem with oblique derivative" ''Mat. Sb.'' , '''87''' : 3 (1972) pp. 417–453 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A. Yanushauskas, ''Dokl. Akad. Nauk SSSR'' , '''164''' : 4 (1965) pp. 753–755  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.I. Vishik, G.I. Eskin, "Sobolev–Slobodinsky spaces of variable order with weighted norm, and their applications to mixed boundary value problems" ''Sibirsk. Mat. Zh.'' , '''9''' : 5 (1968) pp. 973–997 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> G. Giraud, ''Ann. Soc. Math. Polon.'' , '''12''' (1934) pp. 35–54  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian) {{MR|}} {{ZBL|0121.06701}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A.I. Nekrasov, "Exact theory of waves of stationary type on the surface of a heavy fluid" , ''Collected works'' , '''1''' , Moscow (1961) (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) {{MR|0198152}} {{ZBL|0141.08001}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> S.L. Sobolev, ''Mat. Sb.'' , '''2''' : 3 (1937) pp. 465–499  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> S. Agmon, A. Douglis, L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, II." ''Comm. Pure Appl. Math.'' , '''17''' (1964) pp. 35–92 {{MR|162050}} {{ZBL|}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> J. Schauder, ''Math. Z.'' , '''33''' (1931) pp. 602–640  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> J. Leray, J. Schauder, ''Ann. Sci. Ecole Norm. Sup. Ser. 3'' , '''51''' (1934) pp. 45–78  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> R.S. Palais, "Morse theory on Hilbert manifolds" ''Topology'' , '''2''' : 4 (1963) pp. 299–340 {{MR|0158410}} {{ZBL|0122.10702}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) {{MR|0284700}} {{ZBL|0198.14101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) {{MR|0152665}} {{MR|0150320}} {{MR|0138774}} {{ZBL|0127.03505}} {{ZBL|0100.07603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) {{MR|0226183}} {{ZBL|0167.09401}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" ''Uspekhi Mat. Nauk'' : 8 (1941) pp. 171–231 (In Russian) {{MR|}} {{ZBL|0179.43901}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Hörmander, "Pseudo-differential operators and non-elliptic boundary value problems" ''Ann. of Math.'' , '''83''' : 1 (1966) pp. 129–209 {{MR|233064}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.L. Borrelli, "The singular, second order oblique derivative problem" ''J. Math. and Mech.'' , '''16''' : 1 (1966) pp. 51–81 {{MR|0203217}} {{ZBL|0143.14603}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Yu.V. Egorov, V.A. Kondrat'ev, "The oblique derivative problem" ''Mat. Sb.'' , '''78''' : 1 (1969) pp. 148–176 (In Russian) {{MR|0237953}} {{ZBL|0186.43202}} {{ZBL|0165.12202}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.G. Maz'ya, "The degenerate problem with oblique derivative" ''Mat. Sb.'' , '''87''' : 3 (1972) pp. 417–453 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A. Yanushauskas, ''Dokl. Akad. Nauk SSSR'' , '''164''' : 4 (1965) pp. 753–755  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.I. Vishik, G.I. Eskin, "Sobolev–Slobodinsky spaces of variable order with weighted norm, and their applications to mixed boundary value problems" ''Sibirsk. Mat. Zh.'' , '''9''' : 5 (1968) pp. 973–997 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> G. Giraud, ''Ann. Soc. Math. Polon.'' , '''12''' (1934) pp. 35–54  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian) {{MR|}} {{ZBL|0121.06701}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A.I. Nekrasov, "Exact theory of waves of stationary type on the surface of a heavy fluid" , ''Collected works'' , '''1''' , Moscow (1961) (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) {{MR|0198152}} {{ZBL|0141.08001}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> S.L. Sobolev, ''Mat. Sb.'' , '''2''' : 3 (1937) pp. 465–499  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> S. Agmon, A. Douglis, L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, II." ''Comm. Pure Appl. Math.'' , '''17''' (1964) pp. 35–92 {{MR|162050}} {{ZBL|}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> J. Schauder, ''Math. Z.'' , '''33''' (1931) pp. 602–640  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> J. Leray, J. Schauder, ''Ann. Sci. Ecole Norm. Sup. Ser. 3'' , '''51''' (1934) pp. 45–78  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> R.S. Palais, "Morse theory on Hilbert manifolds" ''Topology'' , '''2''' : 4 (1963) pp. 299–340 {{MR|0158410}} {{ZBL|0122.10702}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Garabedian, "Partial differential equations" , Wiley (1964) {{MR|0162045}} {{ZBL|0124.30501}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Friedman, "Partial differential equations" , Holt, Rinehart &amp; Winston (1969) {{MR|0445088}} {{ZBL|0224.35002}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Garabedian, "Partial differential equations" , Wiley (1964) {{MR|0162045}} {{ZBL|0124.30501}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Friedman, "Partial differential equations" , Holt, Rinehart &amp; Winston (1969) {{MR|0445088}} {{ZBL|0224.35002}} </TD></TR></table>

Latest revision as of 06:29, 30 May 2020


The problem of finding a solution $ u $, regular in a domain $ D $, to an elliptic equation

$$ \tag{1 } \sum _ {i, k = 0 } ^ { n } a _ {ik} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {k} } + \sum _ {i = 0 } ^ { n } b _ {i} \frac{\partial u }{\partial x _ {i} } + cu = f, $$

which satisfies certain additional conditions on the boundary $ \Gamma $ of $ D $. Here $ a _ {ik} $, $ b _ {i} $, $ c $ and $ f $ are given functions on $ D $.

The classical boundary value problems are special cases of the following problem: Find a solution to equation (1), regular in a domain $ D $ and satisfying on $ \Gamma $

$$ \tag{2 } a \frac{du }{dl } + bu = g $$

where $ d/dl $ denotes differentiation in some direction, and $ a, b $ and $ g $ are given continuous functions on $ \Gamma $ with $ | a | + | b | > 0 $ everywhere on $ \Gamma $( see [1]).

Putting $ a = 0, b = 1 $, one obtains the Dirichlet problem; with $ b = 0, a = 1 $ one has a problem with oblique derivative (see Differential equation, partial, oblique derivatives), which becomes a Neumann problem if $ l $ is the direction of the conormal. If $ \Gamma = \overline \Gamma \; _ {1} \cup \overline \Gamma \; _ {2} $, where $ \Gamma _ {1} $ and $ \Gamma _ {2} $ are disjoint open subsets of $ \Gamma $, and $ \overline \Gamma \; _ {1} \cap \overline \Gamma \; _ {2} $ is either empty or an $ (n - 2) $- dimensional manifold, with $ a = 1 $, $ b = 0 $ on $ \Gamma _ {1} $, $ a = 0 $, $ b = 1 $ on $ \Gamma _ {2} $, one obtains a mixed problem.

Problem (2) has been studied for elliptic equations in two independent variables (see [2]). Fairly complete investigations have been made of the Dirichlet problem for elliptic equations in any finite number of independent variables (see [1], [3], [4]) and the problem with oblique derivative in case the direction $ l $ is not contained in a tangent plane to $ \Gamma $ at any point of $ \Gamma $. In that case the problem with oblique derivative is a Fredholm problem and the solution is smooth to the same order as the field of directions $ l $ and the function $ g $( see [1]). The case in which $ l $ lies in a tangent plane to $ \Gamma $ at certain points of $ \Gamma $ has been studied (see [3]). The local properties of solutions to the problem with oblique derivative have been investigated (see [5]). At points where the field $ l $ lies in a tangent plane to $ \Gamma $, the solution of the problem is less smooth than $ l $ and $ g $. This has been used as a basis for investigating the problem in a generalized setting (see [7], [8]).

Consider the following boundary problem for harmonic functions regular in the unit ball $ \Sigma \subset \mathbf R ^ {3} $:

$$ au _ {x} + bu _ {y} + cu _ {z} = g; $$

let $ K $ be the set of points of the unit sphere $ S $ at which the function $ \omega = ax + by + cz $ vanishes. The vector field $ P (a, b, c) $ lies in a tangent plane to $ S $ at the points of $ K $. Suppose in addition that $ K $ is the union of a finite number of disjoint curves; let $ K ^ {+} $ be the subset of $ K $ consisting of those points at which $ \mathop{\rm grad} \omega $ makes an acute angle with the projection of the field $ P $ on $ S $, and let $ K ^ {-} $ be the remaining part of $ K $. A generalized formulation of the problem is obtained when the values of $ u $ are also prescribed on $ K ^ {+} $, whereas on $ K ^ {-} $ the solution $ u $ is allowed to have integrable singularities. If $ K ^ {-} $ is empty, the solution to the generalized problem may be made arbitrarily smooth by increasing the smoothness of the additional data of the problem. Generally speaking, a solution to the mixed problem on the set $ \Gamma _ {0} = \overline \Gamma \; _ {1} \cap \overline \Gamma \; _ {2} $ has singularities (see [1]). In order to eliminate such singularities on $ \Gamma _ {0} $, one must impose additional conditions on the data (see [11]).

A large category of boundary value problems is constituted by what are known as problems with free boundaries. In these problems one must find not only a solution of equation (1), but also the domain in which it is regular. The boundary $ \Gamma $ of the domain is unknown, but two boundary conditions must be satisfied on it. An example of this type of problem is the problem of wave motions of an ideal fluid: Find a harmonic function $ u $, regular in some domain $ D $, where part of the boundary, $ \Gamma _ {1} $ say, is known and the normal derivative $ \partial u/ \partial n $ is given on $ \Gamma _ {1} $; the other part of the boundary, $ \Gamma _ {2} $, is unknown and on it one gives two boundary conditions:

$$ \frac{\partial u }{\partial n } = 0,\ \ u _ {x} ^ {2} + u _ {y} ^ {2} + u _ {z} ^ {2} = q (x, y, z), $$

where $ q > 0 $ is a given function.

For harmonic functions of two independent variables, one uses conformal mapping (see [12], [13], [14]). See also Differential equation, partial, free boundaries.

The following problem has been investigated: Find a harmonic function $ u $, regular in a domain $ D $ and satisfying the condition

$$ | \mathop{\rm grad} u | ^ {2} = q, $$

where $ q > 0 $ is a given function, on the boundary $ \Gamma $. There is a complete solution of this problem for harmonic functions of two independent variables (see [14]).

Given an equation $ Lu = f $, where $ L $ is an operator of order $ 2m $, uniformly elliptic in the closure $ \overline{D}\; $ of a domain $ D $, consider the problem of determining a solution $ u $, regular in $ D $ and satisfying on the boundary $ \Gamma $ of $ D $ the conditions

$$ \tag{3 } B _ {j} u = \Phi _ {j} ,\ \ j = 1 \dots m, $$

where $ B _ {j} (x, D), j = 1 \dots m $, are differential operators satisfying the following complementarity condition.

Let $ L ^ \prime (x, \partial / \partial x _ {1} \dots \partial / \partial x _ {n + 1 } ) $ be the principal part of $ L $, let $ B _ {j} ^ { \prime } $ be the principal part of $ B _ {j} $, $ n $ the normal to $ \Gamma $ at a point $ x $ and $ \lambda \neq 0 $ an arbitrary vector parallel to $ \Gamma $. Let $ \tau _ {k} ^ {+} ( \lambda ) $ denote the roots of $ L ^ \prime (x, \lambda + \tau n) $ with positive imaginary parts. The polynomials $ B _ {j} ^ { \prime } (x, \lambda + \tau n) $, $ j = 1 \dots m $, as functions of $ \tau $, must be linearly independent modulo the polynomial $ \prod _ {k = 1 } ^ {m} ( \tau - \tau _ {k} ^ {+} ( \lambda )) $. In this case, too, the problem is normally solvable. Violation of the complementarity condition may entail an essential change in the nature of the problem (see [17]).

Problem (2) is a special case of problem (3). For problem (2) with $ a \equiv 1 $, the complementarity condition is equivalent to the condition that there be no point on the boundary of the domain at which the direction $ l $ lies in a tangent plane to the boundary.

Another particular case of problem (3) is the boundary value problem

$$ \frac{\partial ^ {j} u }{\partial n ^ {j} } = \Phi _ {j} ,\ \ j = 0 \dots m - 1, $$

which is an analogue, to some extent, of the Dirichlet problem for higher-order elliptic equations.

The boundary value problem has been studied for the poly-harmonic equation $ \Delta ^ {k} u = 0 $ when the boundary of the domain consists of manifolds of different dimensions (see [15]).

In investigations of boundary value problems for non-linear equations (e.g. the Dirichlet and Neumann problems), much importance attaches to a priori estimates, various fixed-point principles (see [17], [18]) and the generalization of Morse theory to the infinite-dimensional case (see [19]).

References

[1] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[2] I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) MR0152665 MR0150320 MR0138774 Zbl 0127.03505 Zbl 0100.07603
[3] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) MR0226183 Zbl 0167.09401
[4] M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk : 8 (1941) pp. 171–231 (In Russian) Zbl 0179.43901
[5] L. Hörmander, "Pseudo-differential operators and non-elliptic boundary value problems" Ann. of Math. , 83 : 1 (1966) pp. 129–209 MR233064
[6] R.L. Borrelli, "The singular, second order oblique derivative problem" J. Math. and Mech. , 16 : 1 (1966) pp. 51–81 MR0203217 Zbl 0143.14603
[7] Yu.V. Egorov, V.A. Kondrat'ev, "The oblique derivative problem" Mat. Sb. , 78 : 1 (1969) pp. 148–176 (In Russian) MR0237953 Zbl 0186.43202 Zbl 0165.12202
[8] V.G. Maz'ya, "The degenerate problem with oblique derivative" Mat. Sb. , 87 : 3 (1972) pp. 417–453 (In Russian)
[9] A. Yanushauskas, Dokl. Akad. Nauk SSSR , 164 : 4 (1965) pp. 753–755
[10] M.I. Vishik, G.I. Eskin, "Sobolev–Slobodinsky spaces of variable order with weighted norm, and their applications to mixed boundary value problems" Sibirsk. Mat. Zh. , 9 : 5 (1968) pp. 973–997 (In Russian)
[11] G. Giraud, Ann. Soc. Math. Polon. , 12 (1934) pp. 35–54
[12] M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian) Zbl 0121.06701
[13] A.I. Nekrasov, "Exact theory of waves of stationary type on the surface of a heavy fluid" , Collected works , 1 , Moscow (1961) (In Russian)
[14] F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) MR0198152 Zbl 0141.08001
[15] S.L. Sobolev, Mat. Sb. , 2 : 3 (1937) pp. 465–499
[16] S. Agmon, A. Douglis, L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, II." Comm. Pure Appl. Math. , 17 (1964) pp. 35–92 MR162050
[17] J. Schauder, Math. Z. , 33 (1931) pp. 602–640
[18] J. Leray, J. Schauder, Ann. Sci. Ecole Norm. Sup. Ser. 3 , 51 (1934) pp. 45–78
[19] R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 : 4 (1963) pp. 299–340 MR0158410 Zbl 0122.10702

Comments

References

[a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[a2] P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501
[a3] A. Friedman, "Partial differential equations" , Holt, Rinehart & Winston (1969) MR0445088 Zbl 0224.35002
How to Cite This Entry:
Boundary value problem, elliptic equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_value_problem,_elliptic_equations&oldid=46132
This article was adapted from an original article by A.I. Yanushauskas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article