Resumen de Boundary gradient estimates for solutions of elliptic equations in non-smooth domains
We study the local regularity properties of solutions to the Poisson equation ∆ u = f in Ω near a non-smooth portion of the boundary ∂Ω where the classical Schauder estimates fail. It is shown that if a boundary point x0 can be touched by a ball B ⊂ Ω, then near x0 the derivatives in the tangential directions to ∂Ω at x0 can be estimated independently of the regularity properties of ∂Ω and of the properties of the normal derivatives. The estimates are given in terms of max |f| and the H ̈older quotient of u. We show how the estimates evolve under further assumptions on f. In particular, we derive estimates on |D2 iju| in terms of max |f| and the tangential derivatives of f (the latter need not be bounded at the boundary). The results can be extended to semi-linear equations of the form ∆ u = G(x, u, ∇ u).
