The regularity problem for the Laplace equation in rough domains
The present paper studies the equivalence between solvability of the \(L^{p'}\)-Dirichlet and solvability of the \(L^p\)-regularity problem for the Laplace operator in corkscrew domains \(\Omega\subset \mathbb{R}^{n+1}\), \(n\ge 2\) with uniformly rectifiable boundaries. The main result shows that the Dirichlet problem for the Laplacian with boundary data in \(L^{p'}(\partial \Omega)\) is solvable if and only if the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space \(W^{1,p}(\partial \Omega)\) is solvable, where \(p\in(1,2+\varepsilon)\) and \(1/p +1/p'=1\).\N\NLet us mention that the obtained result also solves the problem posed in [\textit{C. E. Kenig}, Harmonic analysis techniques for second order elliptic boundary value problems: dedicated to the memory of Professor Antoni Zygmund. Providence, RI: American Mathematical Society (1994; Zbl 0812.35001)] and reintroduced in [\textit{T. Toro}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. III: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 1485--1497 (2011; Zbl 1251.28002)] which can be stated as follows:\N\N\textit{In a bounded chord-arc domain \(\Omega \subset \mathbb{R}^{n+1}, n\ge 2\), does there exist \(p > 1\) such that the regularity problem for the Laplacian with boundary data in the Sobolev space \(W^{1,p}(\partial \Omega)\) is solvable? If so, are the layer potentials invertible in appropriate \(L^p\) spaces for such \(p > 1\)?}
scientific article; zbMATH DE number 7875323
scientific article