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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 $$W^{2, p}(\Omega)\cap H^1_0(\Omega)$$.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
##### Keywords:
$$L^ p$$-regularity; domain with cuspidal points