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Neumann and mixed problems on curvilinear polyhedra. (English) Zbl 0767.46026
L’auteur présente un travail approfondi de la régularité des solutions d’un problème mêlé: \(\Delta u=f\), \(u=0\) sur \(\Gamma_ 0\), \({{\partial u} \over {\partial n}}=0\) sur \(\Gamma_ N\) avec \(\overline{\Gamma}_ 0\cup\Gamma_ N=\Gamma\), \(\Gamma_ 0\cap\Gamma_ N=\emptyset\), \(\Gamma\) bord du domaine \(\Omega\) où est définie \(u\), lorsque \(\Omega\) et un polyhèdre curviligne de \(\mathbb{R}^ 2\) ou \(\mathbb{R}^ 3\), de bord \(\Gamma\) admettant des aretes et des sommets.
L’objectif est de préciser les hypothèses pour lesquelles \(u\) solution: \(\int \nabla u\cdot \nabla \overline{v}= \langle f,v\rangle\) \(\forall v\in H^ 1(\Omega)\), \(f\in W^{k,p}(\Omega)\), alors \(u\in W^{k+2,p}(\Omega)\) (théorème 1.1). Elle est en particulier amenée à chercher une estimation de la \(1^ \circ\) valeur propre d’un opérateur de Laplace-Beltrami dans un espace de Sobolev sur un domaine de la sphère \(S^ 2\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J25 Boundary value problems for second-order elliptic equations
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