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The \(L^ p\)-regularity of the solution of the Dirichlet problem in a domain with cuspidal points. (La régularité \(L^ p\) de la solution du problème de Dirichlet dans un domaine à points de rebroussement.) (French) Zbl 0837.35025
Summary: We consider the Dirichlet problem with homogeneous boundary condition in a domain of \(\mathbb{R}^3\) with cuspidal points. We prove that in a domain of the form: \(\Omega= \{(x, y, z)\); \(0< z< a\), \((x/z^\alpha, y/z^\alpha)\in \Omega_0\}\) with \(\Omega_0\) a regular domain of \(\mathbb{R}^2\), \(a> 0\), \(\alpha> 1\) then for any \(f\) in \(L^p(\Omega)\), \(1< p< \infty\), the solution of the Dirichlet problem is in \(W^{2, p}(\Omega)\cap H^1_0(\Omega)\).

35B65 Smoothness and regularity of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation