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A boundary value problem of special form, in which it is required to find solutions to a differential equation
 
A boundary value problem of special form, in which it is required to find solutions to a differential 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/f/f040/f040440/f0404401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
Lu  = f
 +
$$
  
of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404402.png" /> in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404403.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404404.png" /> for given values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404405.png" /> and all its (normal) derivatives of order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404406.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404407.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f0404408.png" /> (or on a part of it). These conditions are usually put in the form
+
of even order $  2m $
 +
in a region $  D $
 +
of the variables $  x = ( x _ {1} \dots x _ {n} ) $
 +
for given values of the function $  u $
 +
and all its (normal) derivatives of order not exceeding $  m- 1 $
 +
on the boundary $  S $
 +
of $  D $(
 +
or on a part of it). These conditions are usually put in 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/f/f040/f040440/f0404409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\left . \left (
 +
\frac \partial {\partial  n }
 +
\right )  ^ {k} u \right | _ {S}  = \phi _ {k} ,\ \
 +
0 \leq  k \leq  m- 1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044010.png" /> is the derivative along the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044011.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044013.png" />, are called the Dirichlet data, and the problem (1), (2) is called a [[Dirichlet problem|Dirichlet problem]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044014.png" />.
+
where $  \partial  / \partial  n $
 +
is the derivative along the outward normal to $  \partial  D $.  
 +
The functions $  \phi _ {k} $,  
 +
0 \leq  k \leq  m- 1 $,  
 +
are called the Dirichlet data, and the problem (1), (2) is called a [[Dirichlet problem|Dirichlet problem]] if $  S = \partial  D $.
  
 
For an ordinary differential equation
 
For an ordinary differential 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/f/f040/f040440/f04044015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
Lu  \equiv  u  ^ {\prime\prime} + a _ {1} u  ^  \prime  + au  = f
 +
$$
  
in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044016.png" /> of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044017.png" />, the first boundary value problem is defined by the boundary conditions
+
in a domain $  D $
 +
of variables $  ( x _ {0} , x _ {1} ) $,  
 +
the first boundary value problem is defined by the boundary 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/f/f040/f040440/f04044018.png" /></td> </tr></table>
+
$$
 +
u( x _ {0} )  = y _ {0} ,\ \
 +
u( x _ {1} )  = y _ {1} .
 +
$$
  
 
For a linear uniformly-elliptic equation
 
For a linear uniformly-elliptic 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/f/f040/f040440/f04044019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
Lu  \equiv  \sum _ {i,j= 1 } ^ { n }  a _ {ij} u _ {x _ {i}  x _ {j} } + \sum _ { i= } 1 ^ { n }  a _ {i} u _ {x _ {i}  } + au  = f
 +
$$
  
 
the first boundary value problem (the Dirichlet problem) consists in finding solutions to this equation subject to the condition
 
the first boundary value problem (the Dirichlet problem) consists in finding solutions to this equation subject to 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/f/f040/f040440/f04044020.png" /></td> </tr></table>
+
$$
 +
\left . u \right | _ {\partial  D }  = \phi .
 +
$$
  
If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044025.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044026.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044027.png" /> are sufficiently smooth, this is a Fredholm problem. In particular, if the measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044028.png" /> is sufficiently small or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044030.png" />, then the problem is uniquely solvable. The smoothness conditions can be weakened considerably not only with respect to the coefficients in the equations and the Dirichlet data but also with respect to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044031.png" />.
+
If the functions $  a _ {ij} $,  
 +
$  a _ {i} $,  
 +
$  a $,  
 +
f $,  
 +
$  \phi $,  
 +
and the $  ( n- 1) $-
 +
dimensional manifold $  \partial  D $
 +
are sufficiently smooth, this is a Fredholm problem. In particular, if the measure of $  D $
 +
is sufficiently small or if $  a \leq  0 $
 +
in $  D $,  
 +
then the problem is uniquely solvable. The smoothness conditions can be weakened considerably not only with respect to the coefficients in the equations and the Dirichlet data but also with respect to the boundary $  \partial  D $.
  
If (1) is a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044032.png" /> equations for an unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044033.png" />-component vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044034.png" />, then the first boundary value problem is posed analogously. In that case there is a substantial difference between the Dirichlet problems for systems (3) and (4): while (3), (2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044035.png" /> is always a Fredholm problem, this needs not apply to the problem (4), (2). For example, the homogeneous Dirichlet problem for the uniformly-elliptic Bitsadze system (cf. [[#References|[1]]])
+
If (1) is a system of $  N > 1 $
 +
equations for an unknown $  N $-
 +
component vector $  u $,  
 +
then the first boundary value problem is posed analogously. In that case there is a substantial difference between the Dirichlet problems for systems (3) and (4): while (3), (2) $  ( S = \partial  D) $
 +
is always a Fredholm problem, this needs not apply to the problem (4), (2). For example, the homogeneous Dirichlet problem for the uniformly-elliptic Bitsadze system (cf. [[#References|[1]]])
  
<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/f/f040/f040440/f04044036.png" /></td> </tr></table>
+
$$
 +
u _ {xx}  ^ {1} - u _ {yy}  ^ {1} - 2u _ {xy}  ^ {2}  = 0,
 +
$$
  
<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/f/f040/f040440/f04044037.png" /></td> </tr></table>
+
$$
 +
2u _ {xy}  ^ {1} + u _ {xx}  ^ {2} - u _ {yy}  ^ {2}  = 0
 +
$$
  
in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044038.png" /> has an infinite number of linearly independent solutions. This example provides a reference point for various additional conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044039.png" /> (regular ellipticity, strong ellipticity) ensuring that the Dirichlet problem is of Fredholm type.
+
in the disc $  x  ^ {2} + y  ^ {2} < R  ^ {2} $
 +
has an infinite number of linearly independent solutions. This example provides a reference point for various additional conditions on $  L $(
 +
regular ellipticity, strong ellipticity) ensuring that the Dirichlet problem is of Fredholm type.
  
 
For linear parabolic equations, the first boundary value problem is posed in a cylinder, and the support for the Dirichlet data is provided by the base and the lateral surface. For example, for the [[Thermal-conductance equation|thermal-conductance equation]] (heat equation)
 
For linear parabolic equations, the first boundary value problem is posed in a cylinder, and the support for the Dirichlet data is provided by the base and the lateral surface. For example, for the [[Thermal-conductance equation|thermal-conductance equation]] (heat 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/f/f040/f040440/f04044040.png" /></td> </tr></table>
+
$$
 +
Lu  \equiv  u _ {t} - \sum _ { i= } 1 ^ { n }  u _ {x _ {i}  x _ {i} }  = 0
 +
$$
  
 
the solution has to be determined in a region
 
the solution has to be determined in a region
  
<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/f/f040/f040440/f04044041.png" /></td> </tr></table>
+
$$
 +
= \{ 0 < t < T, x = ( x _ {1} \dots x _ {n} ) \in G \}
 +
$$
  
and the support for the Dirichlet data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044042.png" /> is
+
and the support for the Dirichlet data $  \phi = u \mid  _ {S} $
 +
is
  
<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/f/f040/f040440/f04044043.png" /></td> </tr></table>
+
$$
 +
= \{ 0 \leq  t \leq  T, x \in \partial  G \} \cup \{ t = 0, x \in \partial
 +
G \} .
 +
$$
  
If the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044044.png" /> is a smooth manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044045.png" />, if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044046.png" /> is smooth and if a compatibility condition is satisfied on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040440/f04044047.png" />, then the first boundary value problem is uniquely solvable.
+
If the boundary $  \partial  G $
 +
is a smooth manifold of dimension $  n- 1 $,  
 +
if the function $  \phi $
 +
is smooth and if a compatibility condition is satisfied on $  \{ t = 0, x \in \partial  G \} $,
 +
then the first boundary value problem is uniquely solvable.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Some classes of partial differential equations" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Hörmander,  "Linear partial differential operators" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Some classes of partial differential equations" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Hörmander,  "Linear partial differential operators" , Springer  (1976)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:39, 5 June 2020


A boundary value problem of special form, in which it is required to find solutions to a differential equation

$$ \tag{1 } Lu = f $$

of even order $ 2m $ in a region $ D $ of the variables $ x = ( x _ {1} \dots x _ {n} ) $ for given values of the function $ u $ and all its (normal) derivatives of order not exceeding $ m- 1 $ on the boundary $ S $ of $ D $( or on a part of it). These conditions are usually put in the form

$$ \tag{2 } \left . \left ( \frac \partial {\partial n } \right ) ^ {k} u \right | _ {S} = \phi _ {k} ,\ \ 0 \leq k \leq m- 1, $$

where $ \partial / \partial n $ is the derivative along the outward normal to $ \partial D $. The functions $ \phi _ {k} $, $ 0 \leq k \leq m- 1 $, are called the Dirichlet data, and the problem (1), (2) is called a Dirichlet problem if $ S = \partial D $.

For an ordinary differential equation

$$ \tag{3 } Lu \equiv u ^ {\prime\prime} + a _ {1} u ^ \prime + au = f $$

in a domain $ D $ of variables $ ( x _ {0} , x _ {1} ) $, the first boundary value problem is defined by the boundary conditions

$$ u( x _ {0} ) = y _ {0} ,\ \ u( x _ {1} ) = y _ {1} . $$

For a linear uniformly-elliptic equation

$$ \tag{4 } Lu \equiv \sum _ {i,j= 1 } ^ { n } a _ {ij} u _ {x _ {i} x _ {j} } + \sum _ { i= } 1 ^ { n } a _ {i} u _ {x _ {i} } + au = f $$

the first boundary value problem (the Dirichlet problem) consists in finding solutions to this equation subject to the condition

$$ \left . u \right | _ {\partial D } = \phi . $$

If the functions $ a _ {ij} $, $ a _ {i} $, $ a $, $ f $, $ \phi $, and the $ ( n- 1) $- dimensional manifold $ \partial D $ are sufficiently smooth, this is a Fredholm problem. In particular, if the measure of $ D $ is sufficiently small or if $ a \leq 0 $ in $ D $, then the problem is uniquely solvable. The smoothness conditions can be weakened considerably not only with respect to the coefficients in the equations and the Dirichlet data but also with respect to the boundary $ \partial D $.

If (1) is a system of $ N > 1 $ equations for an unknown $ N $- component vector $ u $, then the first boundary value problem is posed analogously. In that case there is a substantial difference between the Dirichlet problems for systems (3) and (4): while (3), (2) $ ( S = \partial D) $ is always a Fredholm problem, this needs not apply to the problem (4), (2). For example, the homogeneous Dirichlet problem for the uniformly-elliptic Bitsadze system (cf. [1])

$$ u _ {xx} ^ {1} - u _ {yy} ^ {1} - 2u _ {xy} ^ {2} = 0, $$

$$ 2u _ {xy} ^ {1} + u _ {xx} ^ {2} - u _ {yy} ^ {2} = 0 $$

in the disc $ x ^ {2} + y ^ {2} < R ^ {2} $ has an infinite number of linearly independent solutions. This example provides a reference point for various additional conditions on $ L $( regular ellipticity, strong ellipticity) ensuring that the Dirichlet problem is of Fredholm type.

For linear parabolic equations, the first boundary value problem is posed in a cylinder, and the support for the Dirichlet data is provided by the base and the lateral surface. For example, for the thermal-conductance equation (heat equation)

$$ Lu \equiv u _ {t} - \sum _ { i= } 1 ^ { n } u _ {x _ {i} x _ {i} } = 0 $$

the solution has to be determined in a region

$$ D = \{ 0 < t < T, x = ( x _ {1} \dots x _ {n} ) \in G \} $$

and the support for the Dirichlet data $ \phi = u \mid _ {S} $ is

$$ S = \{ 0 \leq t \leq T, x \in \partial G \} \cup \{ t = 0, x \in \partial G \} . $$

If the boundary $ \partial G $ is a smooth manifold of dimension $ n- 1 $, if the function $ \phi $ is smooth and if a compatibility condition is satisfied on $ \{ t = 0, x \in \partial G \} $, then the first boundary value problem is uniquely solvable.

References

[1] A.V. Bitsadze, "Some classes of partial differential equations" , Gordon & Breach (1988) (Translated from Russian)
[2] L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
[3] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[4] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[5] I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)
[6] A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian)
[7] L. Hörmander, "Linear partial differential operators" , Springer (1976)

Comments

The first boundary value problem has been generalized to the notion of an elliptic boundary value problem. See [a1], Chapt. 20, also for a study of the Fredholm properties.

For the theory of boundary value problems for non-linear elliptic equations, which undergoes intensive development nowadays (1988), see [a2], [a3].

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)
[a2] D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1977)
[a3] O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)
[a4] A. Friedman, "Partial differential equations" , Holt, Rinehart & Winston (1969)
[a5] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[a6] P.R. Garabedian, "Partial differential equations" , Wiley (1964)
How to Cite This Entry:
First boundary value problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=First_boundary_value_problem&oldid=46933
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article