Schauder estimates explained

In mathematics, and more precisely, in functional Analysis and PDEs, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates.

There is both an interior result, giving a Hölder condition for the solution in interior domains away from the boundary, and a boundary result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well.

The Schauder estimates are a necessary precondition to using the method of continuity to prove the existence and regularity of solutions to the Dirichlet problem for elliptic PDEs. This result says that when the coefficients of the equation and the nature of the boundary conditions are sufficiently smooth, there is a smooth classical solution to the PDE.

Notation

The Schauder estimates are given in terms of weighted Hölder norms; the notation will follow that given in the text of .

The supremum norm of a continuous function

f\inC(\Omega)

is given by

|f|_0;\Omega=\sup_x\in\Omega|f(x)|

For a function which is Hölder continuous with exponent

\alpha

, that is to say

f\inC^0,(\Omega)

, the usual Hölder seminorm is given by

[f]_{0,\alpha;\Omega}=\sup_x,y\in

	\|f(x)-f(y)\|
	\|x-y\|^\alpha

The sum of the two is the full Hölder norm of f

|f|_{0,\alpha;\Omega}=|f|_0;\Omega+[f]_{0,\alpha;\Omega}=\sup_x\in\Omega|f(x)|+\sup_x,y\in

	\|f(x)-f(y)\|
	\|x-y\|^\alpha

For differentiable functions u, it is necessary to consider the higher order norms, involving derivatives. The norm in the space of functions with k continuous derivatives,

C^k(\Omega)

, is given by

where

\beta

ranges over all multi-indices of appropriate orders. For functions with kth order derivatives which are Hölder continuous with exponent

\alpha

, the appropriate semi-norm is given by

[u]_{k,\alpha;\Omega}=\sup_{\stackrel{x,y\in}{|\beta|=k}}

	\|D^\betau(x)-D^\betau(y)\|
	\|x-y\|^\alpha

which gives a full norm of

|u|_{k,\alpha;\Omega}=|u|_k;\Omega+[u]_{k,\alpha;\Omega}=\sum_|\beta|\sup_x\in|D^\betau(x)|+\sup_{\stackrel{x,y\in}{|\beta|=k}}

	\|D^\betau(x)-D^\betau(y)\|
	\|x-y\|^\alpha

For the interior estimates, the norms are weighted by the distance to the boundary

d_x=d(x,\partial\Omega)

raised to the same power as the derivative, and the seminorms are weighted by

d_x,y=min(d_x,d_y)

raised to the appropriate power. The resulting weighted interior norm for a function is given by

	*
\|u\|
	k,\alpha;\Omega

	*
\|u\|
	k;\Omega

	*
[u]
	k,\alpha;\Omega

=\sum_|\beta|\sup_x\in

	\|\beta\|
\|d
	x

D^\betau(x)|+\sup_{\stackrel{x,y\in}{|\beta|=k}}

	k+\alpha
d
	x,y

	\|D^\betau(x)-D^\betau(y)\|
	\|x-y\|^\alpha

It is occasionally necessary to add "extra" powers of the weight, denoted by

	(m)
\|u\|
	k,\alpha;\Omega

	(m)
\|u\|
	k;\Omega

	(m)
[u]
	k,\alpha;\Omega

=\sum_|\beta|\sup_x\in

	\|\beta\|+m
\|d
	x

D^\betau(x)|+\sup_{\stackrel{x,y\in}{|\beta|=k}}

	m+k+\alpha
d
	x,y

	\|D^\betau(x)-D^\betau(y)\|
	\|x-y\|^\alpha

Formulation

The formulations in this section are taken from the text of .

Interior estimates

Consider a bounded solution

u\inC^2,\alpha(\Omega)

on the domain

\Omega

to the elliptic, second order, partial differential equation

\sum_i,ja_i,j(x)D_iD_ju(x)+\sum_ib_i(x)D_iu(x)+c(x)u(x)=f(x)

where the source term satisfies

f\inC^{\alpha(\Omega)}

. If there exists a constant

λ>0

such that the

a_i,j

are strictly elliptic,

\suma_i,j(x)\xi_i\xi_j\geqλ|\xi|²

for all

x\in\Omega,\xi\inRⁿ

and the relevant norms coefficients are all bounded by another constant

|a_i,j|_{0,\alpha;\Omega},

	(1)
\|b
	0,\alpha;\Omega

	(2)
\|c\|
	0,\alpha;\Omega

\leqΛ.

Then the weighted

C^2,\alpha

norm of u is controlled by the supremum of u and the Hölder norm of f:

	*
\|u\|
	2,\alpha;\Omega

\leqC(n,\alpha,λ,Λ)(|u|_0,\Omega+

	(2)
\|f\|
	0,\alpha;\Omega

Boundary estimates

Let

\Omega

be a

C^2,\alpha

domain (that is to say, about any point on the boundary of the domain the boundary hypersurface can be realized, after an appropriate rotation of coordinates, as a

C^2,\alpha

function), with Dirichlet boundary data that coincides with a function

\phi(x)

which is also at least

C^2,\alpha

. Then subject to analogous conditions on the coefficients as in the case of the interior estimate, the unweighted Hölder norm of u is controlled by the unweighted norms of the source term, the boundary data, and the supremum norm of u:

|u|_{2,\alpha;\Omega}\leqC(n,\alpha,λ,Λ,\Omega)(|u|_0,\Omega+|f|_{0,\alpha;\Omega}+|\phi|_{2,\alpha;\partial\Omega}).

When the solution u satisfies the maximum principle, the first factor on the right hand side can be dropped.