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Problèmes mixtes pour le laplacien dans des domaines polyédraux courbes. (Mixed problems for the Laplace operator on polyhedral curvilinear domains). (French) Zbl 0715.35022
This note presents in a condensed manner some regularity results of weak solutions of the Laplacian. The underlying class of domains $$\Omega$$ consists of three-dimensional cylindrical ones: $\Omega =\{(y,z)\in {\mathbb{R}}\times {\mathbb{R}}^ 2| \quad z\in P,\quad \Phi_ 0(z)<y<\Phi_ 1(z)\}.$ Here P is a curvilinear polygon in $${\mathbb{R}}^ 2$$ with “sides” being curves of class $$C^{\rho}$$, $$\rho\geq 1$$. $$\Phi_ 0$$ and $$\Phi_ 1$$ are functions of class $$C^{\rho}(P)$$. It is assumed that P does not contain cusp points. However, fissures are admitted, what is important from the point of view of applications to problems in solid mechanics. Main results formulated in the paper concern regularity of weak solutions for the Dirichlet, Neumann and mixed problems, provided that $$f\in W_ p^{s-1}(\Omega)$$ or $$f\in W^ 1_ q(\Omega)'$$. As usual, f denotes the right hand side of the equation.
Another result concerns existence of a solution for the following problem $w\in W^ 1_ p(\Omega,{\mathbb{R}}^ 3),\quad curl \underline w=\underline g,\quad div \underline w=0,\quad \underline w\cdot \underline n|_{\Gamma_ 1}=0,\quad \underline w\times \underline n|_{\Gamma_ 2}=0,$ provided that \bg$$\in L^ p(\Omega,{\mathbb{R}}^ 3).$$
This valuable contribution will be of interest not only to specialists in P.D.E., but also to mathematically oriented mechanicians.
Reviewer: J.J.Telega

##### MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35D10 Regularity of generalized solutions of PDE (MSC2000) 35F15 Boundary value problems for linear first-order PDEs 35Q60 PDEs in connection with optics and electromagnetic theory
##### Keywords:
Maxwell equations; Dirichlet; Neumann; mixed problems; existence