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
![]() | (1) |
given in a bounded domain of a Euclidean space of points
and with a coefficient
, can be described in the following way.
1) The spaces ,
and
are introduced as sets of functions
with finite norms
![]() |
![]() |
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2) It is assumed that the boundary of the domain
is of class
, i.e. each element
of the
-dimensional surface
can be mapped on a part of the plane by a coordinate transformation
with a positive Jacobian, moreover,
.
3) It is proved that if the coefficients of (1) belong to the space and if the function
, then the a priori estimate
![]() | (2) |
is true up to the boundary, where the constant depends only on
, on the ellipticity constant
,
, and on the norms of the coefficients of the operator
, and where
![]() |
4) It is assumed that one knows how to prove the existence of a solution to the Dirichlet problem
![]() |
for the Laplace operator .
5) Without loss of generality one may assume that , and then apply the continuation method, the essence of which is the following:
. The operator
is imbedded in a one-parameter family of operators
![]() |
. Basing oneself essentially on the a priori estimate (2), it can be established that the set
of those values of
for which the Dirichlet problem
,
, has a solution
for all
, is at the same time open and closed, and thus coincides with the unit interval
.
6) It is proved that if is a bounded domain contained in
together with its closure, then for any function
and any compact subdomain
the interior a priori estimate
![]() | (3) |
holds.
7) Approximating uniformly the functions and
by functions from
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
, with boundaries of the same smoothness as
.
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) |
Schauder method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schauder_method&oldid=48617