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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)]]).

Schauder's method of finding a solution to the [[Dirichlet problem|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 [[#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.

====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>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gilbarg,   N.S. Trudinger,   "Elliptic partial differential equations of second order" , Springer  (1977)</TD></TR></table>