In mathematics, the Schauder estimates are a collection of results due to Juliusz Schauder (1934, 1937) 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 do not assume the existence of the solution, they are called a priori estimates.
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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 proving 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 D. Gilbarg and Neil Trudinger (1983).
The supremum norm of a continuous function
For a function which is Hölder continuous with exponent
The sum of the two is the full Hölder norm of f
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,
where
which gives a full norm of
For the interior estimates, the norms are weighted by the distance to the boundary
raised to the same power as the derivative, and the seminorms are weighted by
raised to the appropriate power. The resulting weighted interior norm for a function is given by
It is occasionally necessary to add "extra" powers of the weight, denoted by
Formulation
The formulations in this section are taken from the text of D. Gilbarg and Neil Trudinger (1983).
Interior estimates
Consider a bounded solution
where the source term satisfies
and the relevant norms coefficients are all bounded by another constant
Then the weighted
Boundary estimates
Let
When the solution u satisfies the maximum principle, the first term on the right hand side can be dropped.