Schauder estimates  In mathematics, 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 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. 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 formula_1 is given by For a function which is Hölder continuous with exponent formula_3, that is to say, formula_4 the usual Hölder seminorm is given by  The formulations in this section are taken from the text of . Consider a bounded solution formula_6 on the domain formula_7 to the elliptic, second order partial differential equation  where the source term satisfies formula_9. If there exists a constant formula_10 such that the formula_11 are strictly elliptic,  and the relevant norms coefficients are all bounded by another constant formula_14  Then the weighted formula_16 norm of "u" is controlled by the supremum of "u" and the Holder norm of "f":  Let formula_7 be a formula_16 domain (that is to say, about any point on the boundary of the domain the boundary surface can be realized, after an appropriate rotation of coordinates, as a formula_16 function), with Dirichlet boundary data that coincides with a function formula_21 which is also at least formula_16. Then subject to analogous conditions on the coefficients as in the case of the interior estimate, the unweighted Holder norm of "u" is controlled by the unweighted norms of the source term, the boundary data, and the supremum norm of "u":  When the solution "u" satisfies the maximum principle, the first term on the right hand side can be dropped.