Schauder method

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)