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A method for solving boundary value problems for linear uniformly-elliptic equations of the second order, based on a priori estimates and the continuation method (see also [[Continuation method (to a parametrized family)|Continuation method (to a parametrized family)]]).
 
A method for solving boundary value problems for linear uniformly-elliptic equations of the second order, based on a priori estimates and the continuation method (see also [[Continuation method (to a parametrized family)|Continuation method (to a parametrized family)]]).
  
 
Schauder's method of finding a solution to the [[Dirichlet problem|Dirichlet problem]] for a linear uniformly-elliptic equation
 
Schauder's method of finding a solution to the [[Dirichlet problem|Dirichlet problem]] 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/s/s083/s083310/s0833101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
Lu  \equiv  \sum _ {i, j= 1 } ^ { n }
 +
a  ^ {ij} ( x) u _ {x _ {i}  x _ {j} } +
 +
\sum _ { j= } 1 ^ { n }  b  ^ {j} ( x) u _ {x _ {j}  } + b
 +
( x) u  = f( x),
 +
$$
  
given in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833102.png" /> of a Euclidean space of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833103.png" /> and with a coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833104.png" />, can be described in the following way.
+
given in a bounded domain $  \Omega $
 +
of a Euclidean space of points $  x= ( x _ {1} \dots x _ {n} ) $
 +
and with a coefficient $  b( x) \leq  0 $,  
 +
can be described in the following way.
  
1) The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833107.png" /> are introduced as sets of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s0833108.png" /> with finite norms
+
1) The spaces $  C _  \alpha  ( \Omega ) $,  
 +
$  C _ {1+ \alpha }  ( \Omega ) $
 +
and $  C _ {2+ \alpha }  ( \Omega ) $
 +
are introduced as sets of functions $  u = u( x) $
 +
with finite norms
  
<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/s/s083/s083310/s0833109.png" /></td> </tr></table>
+
$$
 +
\| u \| _ {C _  \alpha  ( \Omega ) }  = \sup _ {x \in \Omega }  | u( x) | +
 +
\sup _ {x,y } 
 +
\frac{u( x)- u( y) }{| x- y |  ^  \alpha  }
 +
,\ \
 +
0 < \alpha < 1,
 +
$$
  
<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/s/s083/s083310/s08331010.png" /></td> </tr></table>
+
$$
 +
\| u \| _ {C _ {1+ \alpha }  ( \Omega ) }  = \| u \| _ {C _  \alpha  (
 +
\Omega ) } + \sum _ { i= } 1 ^ { n }  \| u _ {x _ {i}  } \| _ {C _  \alpha  ( \Omega ) } ,
 +
$$
  
<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/s/s083/s083310/s08331011.png" /></td> </tr></table>
+
$$
 +
\| u \| _ {C _ {2+ \alpha }  ( \Omega ) }  = \| u \| _ {C _ {1+ \alpha }
 +
( \Omega ) } + \sum _ { i,j= } 1 ^ { n }  \| u _ {x _ {i}  x _ {j} } \| _ {C _  \alpha  ( \Omega ) } .
 +
$$
  
2) It is assumed that the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331012.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331013.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331014.png" />, i.e. each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331015.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331016.png" />-dimensional surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331017.png" /> can be mapped on a part of the plane by a coordinate transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331018.png" /> with a positive Jacobian, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331019.png" />.
+
2) It is assumed that the boundary $  \sigma $
 +
of the domain $  \Omega $
 +
is of class $  C _ {2 + \alpha }  $,  
 +
i.e. each element $  \sigma _ {x} $
 +
of the $  ( n- 1) $-
 +
dimensional surface $  \sigma $
 +
can be mapped on a part of the plane by a coordinate transformation $  y= y( x) $
 +
with a positive Jacobian, moreover, $  u \in C _ {2 + \alpha }  ( \sigma _ {x} ) $.
  
3) It is proved that if the coefficients of (1) belong to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331020.png" /> and if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331021.png" />, then the a priori estimate
+
3) It is proved that if the coefficients of (1) belong to the space $  C _  \alpha  ( \Omega ) $
 +
and if the function $  u \in C _ {2+ \alpha }  ( \Omega ) $,  
 +
then the a priori estimate
  
<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/s/s083/s083310/s08331022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\| u \| _ {C _ {2+ \alpha }  ( \Omega ) }  \leq  C  \left [ \| Lu \| _ {C _  \alpha  ( \Omega ) }  + \| u \| _ {C _ {2+ \alpha }  ( \Omega ) } + \| u \| _ {C _ {0(
 +
\Omega ) } \right ]
 +
$$
  
is true up to the boundary, where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331023.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331024.png" />, on the ellipticity constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331026.png" />, and on the norms of the coefficients of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331027.png" />, and where
+
is true up to the boundary, where the constant $  C $
 +
depends only on $  \Omega $,  
 +
on the ellipticity constant $  m \leq  a  ^ {ij} ( x) \xi _ {i} \xi _ {j} / | \xi |  ^ {2} $,  
 +
$  \xi \neq 0 $,  
 +
and on the norms of the coefficients of the operator $  L $,  
 +
and where
  
<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/s/s083/s083310/s08331028.png" /></td> </tr></table>
+
$$
 +
\| u \| _ {C _ {0}  ( \Omega ) }  = \sup _ {x \in \Omega }  | u( x) | .
 +
$$
  
4) It is assumed that one knows how to prove the existence of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331029.png" /> to the Dirichlet problem
+
4) It is assumed that one knows how to prove the existence of a solution $  u \in C _ {2+ \alpha }  $
 +
to the Dirichlet 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/s/s083/s083310/s08331030.png" /></td> </tr></table>
+
$$
 +
\left . u \right | _  \sigma  = \left . \phi \right | _  \sigma  ,\ \
 +
\phi \in C _ {2+ \alpha }  ( \Omega ) ,
 +
$$
  
for the Laplace operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331031.png" />.
+
for the Laplace operator $  \Delta = \sum _ {i=} 1  ^ {n} \partial  ^ {2} / \partial  x _ {i}  ^ {2} $.
  
5) Without loss of generality one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331032.png" />, and then apply the continuation method, the essence of which is the following:
+
5) Without loss of generality one may assume that $  \phi ( x) \equiv 0 $,  
 +
and then apply the continuation method, the essence of which is the following:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331033.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331034.png" /> is imbedded in a one-parameter family of operators
+
$  5 _ {1} $.  
 +
The operator $  L $
 +
is imbedded in a one-parameter family of operators
  
<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/s/s083/s083310/s08331035.png" /></td> </tr></table>
+
$$
 +
L _ {t} u  = tLu + ( 1- t) \Delta u ,\ \
 +
0 \leq  t \leq  1,\ \
 +
L _ {0= \Delta .
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331036.png" />. Basing oneself essentially on the a priori estimate (2), it can be established that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331037.png" /> of those values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331038.png" /> for which the Dirichlet problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331040.png" />, has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331041.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331042.png" />, is at the same time open and closed, and thus coincides with the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331043.png" />.
+
$  5 _ {2} $.  
 +
Basing oneself essentially on the a priori estimate (2), it can be established that the set $  T $
 +
of those values of $  t \in [ 0, 1] $
 +
for which the Dirichlet problem $  L _ {t} u = f( x) $,  
 +
$  u \mid  _  \sigma  = 0 $,  
 +
has a solution $  u \in C _ {2+ \alpha }  ( \Omega ) $
 +
for all $  f \in C _  \alpha  ( \Omega ) $,  
 +
is at the same time open and closed, and thus coincides with the unit interval $  [ 0, 1] $.
  
6) It is proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331044.png" /> is a bounded domain contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331045.png" /> together with its closure, then for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331046.png" /> and any compact subdomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331047.png" /> the interior a priori estimate
+
6) It is proved that if $  D $
 +
is a bounded domain contained in $  \Omega $
 +
together with its closure, then for any function $  u \in C _ {2+ \alpha }  ( D) $
 +
and any compact subdomain $  \omega \subset  D $
 +
the interior a priori estimate
  
<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/s/s083/s083310/s08331048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\| u \| _ {C _ {2+ \alpha }  ( \omega ) }  \leq  C  \left [ \| Lu \| _ {C _  \alpha  ( D) }  + \| u \| _ {C _ {0( D) } \right ]
 +
$$
  
 
holds.
 
holds.
  
7) Approximating uniformly the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331050.png" /> by functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331051.png" /> and applying the estimate (3), one proves the existence of a solution to the Dirichlet problem for any continuous boundary function and for a wide class of domains with non-smooth boundaries, e.g. for domains that can be represented as the union of sequences of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331052.png" />, with boundaries of the same smoothness as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083310/s08331053.png" />.
+
7) Approximating uniformly the functions $  \phi $
 +
and $  f $
 +
by functions from $  C _ {2+ \alpha }  $
 +
and applying the estimate (3), one proves the existence of a solution to the Dirichlet problem for any continuous boundary function and for a wide class of domains with non-smooth boundaries, e.g. for domains that can be represented as the union of sequences of domains $  \Omega _ {1} \subset  \Omega _ {2} \subset  \dots $,  
 +
with boundaries of the same smoothness as $  \sigma $.
  
 
Estimates 2 and 3 where first obtained by J. Schauder (see [[#References|[1]]], [[#References|[2]]]) and go under his name. Schauder's estimates and his method have been generalized to equations and systems of higher order. The a priori estimates, both interior and up to the boundary, corresponding to it are sometimes called Schauder-type estimates. The method of a priori estimates is a further generalization of Schauder's method.
 
Estimates 2 and 3 where first obtained by J. Schauder (see [[#References|[1]]], [[#References|[2]]]) and go under his name. Schauder's estimates and his method have been generalized to equations and systems of higher order. The a priori estimates, both interior and up to the boundary, corresponding to it are sometimes called Schauder-type estimates. The method of a priori estimates is a further generalization of Schauder's method.
Line 51: Line 134:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Schauder,  "Ueber lineare elliptische Differentialgleichungen zweiter Ordnung"  ''Math. Z.'' , '''38''' :  2  (1934)  pp. 257–282</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Schauder,  "Numerische Abschätzungen in elliptischen linearen Differentialgleichungen"  ''Studia Math.'' , '''5'''  (1935)  pp. 34–42</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[4]</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">[5]</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">[6]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,  "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  O.A. Ladyzhenskaya,  N.N. Ural'tseva,  "Linear and quasilinear elliptic equations" , Acad. Press  (1968)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Schauder,  "Ueber lineare elliptische Differentialgleichungen zweiter Ordnung"  ''Math. Z.'' , '''38''' :  2  (1934)  pp. 257–282</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Schauder,  "Numerische Abschätzungen in elliptischen linearen Differentialgleichungen"  ''Studia Math.'' , '''5'''  (1935)  pp. 34–42</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[4]</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">[5]</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">[6]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,  "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  O.A. Ladyzhenskaya,  N.N. Ural'tseva,  "Linear and quasilinear elliptic equations" , Acad. Press  (1968)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:12, 6 June 2020


A method for solving boundary value problems for linear uniformly-elliptic equations of the second order, based on a priori estimates and the continuation method (see also Continuation method (to a parametrized family)).

Schauder's method of finding a solution to the Dirichlet problem for a linear uniformly-elliptic equation

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

given in a bounded domain $ \Omega $ of a Euclidean space of points $ x= ( x _ {1} \dots x _ {n} ) $ and with a coefficient $ b( x) \leq 0 $, can be described in the following way.

1) The spaces $ C _ \alpha ( \Omega ) $, $ C _ {1+ \alpha } ( \Omega ) $ and $ C _ {2+ \alpha } ( \Omega ) $ are introduced as sets of functions $ u = u( x) $ with finite norms

$$ \| u \| _ {C _ \alpha ( \Omega ) } = \sup _ {x \in \Omega } | u( x) | + \sup _ {x,y } \frac{u( x)- u( y) }{| x- y | ^ \alpha } ,\ \ 0 < \alpha < 1, $$

$$ \| u \| _ {C _ {1+ \alpha } ( \Omega ) } = \| u \| _ {C _ \alpha ( \Omega ) } + \sum _ { i= } 1 ^ { n } \| u _ {x _ {i} } \| _ {C _ \alpha ( \Omega ) } , $$

$$ \| u \| _ {C _ {2+ \alpha } ( \Omega ) } = \| u \| _ {C _ {1+ \alpha } ( \Omega ) } + \sum _ { i,j= } 1 ^ { n } \| u _ {x _ {i} x _ {j} } \| _ {C _ \alpha ( \Omega ) } . $$

2) It is assumed that the boundary $ \sigma $ of the domain $ \Omega $ is of class $ C _ {2 + \alpha } $, i.e. each element $ \sigma _ {x} $ of the $ ( n- 1) $- dimensional surface $ \sigma $ can be mapped on a part of the plane by a coordinate transformation $ y= y( x) $ with a positive Jacobian, moreover, $ u \in C _ {2 + \alpha } ( \sigma _ {x} ) $.

3) It is proved that if the coefficients of (1) belong to the space $ C _ \alpha ( \Omega ) $ and if the function $ u \in C _ {2+ \alpha } ( \Omega ) $, then the a priori estimate

$$ \tag{2 } \| u \| _ {C _ {2+ \alpha } ( \Omega ) } \leq C \left [ \| Lu \| _ {C _ \alpha ( \Omega ) } + \| u \| _ {C _ {2+ \alpha } ( \Omega ) } + \| u \| _ {C _ {0} ( \Omega ) } \right ] $$

is true up to the boundary, where the constant $ C $ depends only on $ \Omega $, on the ellipticity constant $ m \leq a ^ {ij} ( x) \xi _ {i} \xi _ {j} / | \xi | ^ {2} $, $ \xi \neq 0 $, and on the norms of the coefficients of the operator $ L $, and where

$$ \| u \| _ {C _ {0} ( \Omega ) } = \sup _ {x \in \Omega } | u( x) | . $$

4) It is assumed that one knows how to prove the existence of a solution $ u \in C _ {2+ \alpha } $ to the Dirichlet problem

$$ \left . u \right | _ \sigma = \left . \phi \right | _ \sigma ,\ \ \phi \in C _ {2+ \alpha } ( \Omega ) , $$

for the Laplace operator $ \Delta = \sum _ {i=} 1 ^ {n} \partial ^ {2} / \partial x _ {i} ^ {2} $.

5) Without loss of generality one may assume that $ \phi ( x) \equiv 0 $, and then apply the continuation method, the essence of which is the following:

$ 5 _ {1} $. The operator $ L $ is imbedded in a one-parameter family of operators

$$ L _ {t} u = tLu + ( 1- t) \Delta u ,\ \ 0 \leq t \leq 1,\ \ L _ {0} = \Delta . $$

$ 5 _ {2} $. Basing oneself essentially on the a priori estimate (2), it can be established that the set $ T $ of those values of $ t \in [ 0, 1] $ for which the Dirichlet problem $ L _ {t} u = f( x) $, $ u \mid _ \sigma = 0 $, has a solution $ u \in C _ {2+ \alpha } ( \Omega ) $ for all $ f \in C _ \alpha ( \Omega ) $, is at the same time open and closed, and thus coincides with the unit interval $ [ 0, 1] $.

6) It is proved that if $ D $ is a bounded domain contained in $ \Omega $ together with its closure, then for any function $ u \in C _ {2+ \alpha } ( D) $ and any compact subdomain $ \omega \subset D $ the interior a priori estimate

$$ \tag{3 } \| u \| _ {C _ {2+ \alpha } ( \omega ) } \leq C \left [ \| Lu \| _ {C _ \alpha ( D) } + \| u \| _ {C _ {0} ( D) } \right ] $$

holds.

7) Approximating uniformly the functions $ \phi $ and $ f $ by functions from $ C _ {2+ \alpha } $ and applying the estimate (3), one proves the existence of a solution to the Dirichlet problem for any continuous boundary function and for a wide class of domains with non-smooth boundaries, e.g. for domains that can be represented as the union of sequences of domains $ \Omega _ {1} \subset \Omega _ {2} \subset \dots $, with boundaries of the same smoothness as $ \sigma $.

Estimates 2 and 3 where first obtained by J. Schauder (see [1], [2]) and go under his name. Schauder's estimates and his method have been generalized to equations and systems of higher order. The a priori estimates, both interior and up to the boundary, corresponding to it are sometimes called Schauder-type estimates. The method of a priori estimates is a further generalization of Schauder's method.

References

[1] J. Schauder, "Ueber lineare elliptische Differentialgleichungen zweiter Ordnung" Math. Z. , 38 : 2 (1934) pp. 257–282
[2] J. Schauder, "Numerische Abschätzungen in elliptischen linearen Differentialgleichungen" Studia Math. , 5 (1935) pp. 34–42
[3] L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
[4] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[5] A.V. Bitsadze, "Some classes of partial differential equations" , Gordon & Breach (1988) (Translated from Russian)
[6] Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)
[7] O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)

Comments

Schauder-type estimates for parabolic equations were obtained for the first time in [a1] (see also [a2] for a detailed description).

References

[a1] C. Ciliberto, "Formule di maggiorazione e teoremi di esistenza per le soluzioni delle equazioni paraboliche in due variabili" Ricerche Mat. , 3 (1954) pp. 40–75
[a2] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[a3] D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1977)
How to Cite This Entry:
Schauder method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schauder_method&oldid=48617
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article