Weighted a priori estimate in straightenable domains of local Lyapunov-Dini type

A. I. Parfenov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 65-150.

Résumé

We prove the well-posedness of the Dirichlet problem for the Poisson equation in a weighted Sobolev space under weak assumptions both on the weight and on the boundary of the domain. The weight is supposed to satisfy the Muckenhoupt condition on the off-boundary cubes and an additional condition near the boundary. The boundary is Lipschitz, flat enough, straightenable (in a sense close to the one studied before by the author) and is either straightenable with small constant or satisfies the so-called local Lyapunov-Dini condition. The proof amounts to an a priori estimate obtained via localizing the problem, straightening the boundary, $L^p_w$-discretizing singular integrals and estimating a number of dyadic sums. Our results strengthen some of the results of V. G. Maz'ya, T. O. Shaposhnikova, K. Schumacher, R. G. Durán, M. Sanmartino and M. Toschi.

Détail

Mots-clés : Poisson equation, dyadic cube.
Keywords: weighted Sobolev space, Muckenhoupt weight, power weight, Lyapunov-Dini domain, straightenable domain, pointwise multiplier, discretization

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