Geodesic
Weighted a priori estimate in straightenable domains of local Lyapunov-Dini type
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 65-150

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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.
Mots-clés : Poisson equation, dyadic cube.
Keywords: weighted Sobolev space, Muckenhoupt weight, power weight, Lyapunov-Dini domain, straightenable domain, pointwise multiplier, discretization 
@article{SEMR_2012_9_a23,
     author = {A. I. Parfenov},
     title = {Weighted a priori estimate in straightenable domains of local {Lyapunov-Dini} type},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {65--150},
     publisher = {mathdoc},
     volume = {9},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2012_9_a23/}
}
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A. I. Parfenov. Weighted a priori estimate in straightenable domains of local Lyapunov-Dini type. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 65-150. http://geodesic.mathdoc.fr/item/SEMR_2012_9_a23/