Boundary gradient estimates for solutions of elliptic equations in non-smooth domains
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[1] Universidad de Oviedo
Universidad de Oviedo
Oviedo, España
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- Localización: XVII Congreso de Ecuaciones Diferenciales y Aplicaciones ; VII Congreso de Matemática Aplicada: Salamanca, 14-28 septiembre 2001 / coord. por Luis Ferragut Canals
, Anastasio Pedro Santos Yanguas , 2001, ISBN 8469961446, págs. 665-666 - Idioma: inglés
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Resumen
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).
