Steux, Jean-Luc The Dirichlet problem for an elliptic operator in a domain with cusp point. (Problème de Dirichlet pour un opérateur elliptique dans un domaine à point cuspide.) (French) Zbl 0939.35060 Ann. Fac. Sci. Toulouse, VI. Sér., Math. 6, No. 1, 143-175 (1997). The paper is devoted to the study of qualitative properties of solutions to the homogeneous Dirichlet boundary value problem for a properly elliptic operator of the order \(2m\) with smooth coefficients on a plane domain \(U\) with a boundary having a cusp point in the origin \(0\). The boundary of \(U\) in a sufficiently small neighbourhood of \(0\) is described as a graph of two smooth functions \(\varphi_1, \varphi_2\) such that \[ \varphi_1, \varphi_2 \in C^1([0,\epsilon]) \cap C^{\infty}(]0,\epsilon]), \]\[ \varphi_1 < \varphi_2,\quad \varphi_1(0) = \varphi_2(0) = 0,\quad \varphi_1'(0) = \varphi_2'(0) = 0. \] The author gives conditions on \(\varphi_1, \varphi_2\) guaranteeing the smoothness of solutions or eigenfunctions of the problem and, if a solution is not smooth, he gives an asymptotic expansion near the cusp point. Reviewer: J.Stará (Praha) Cited in 2 Documents MSC: 35J40 Boundary value problems for higher-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs Keywords:Dirichlet problem; properly elliptic operator; plane domain with a cusp point; asymptotics of a solution; regularity of solutions PDFBibTeX XMLCite \textit{J.-L. Steux}, Ann. Fac. Sci. Toulouse, Math. (6) 6, No. 1, 143--175 (1997; Zbl 0939.35060) Full Text: DOI Numdam EuDML References: [1] Agmon, S.), Douglis, A.) et Nirenberg, L.) .- Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math.17 (1964), pp. 35-92. · Zbl 0123.28706 [2] Dauge, M.) .- Elliptic Boundary Value Problems on Corner Domains, , Springer-Verlag1341 (1988). · Zbl 0668.35001 [3] Dauge, M.) et Steux, J.-L.) .- Problème de Dirichlet pour le laplacien dans un polygône curviligne, Journal of Differential Equations70, n° 1 (1987), pp. 93-113. · Zbl 0642.35064 [4] Grisvard, P.) .- Elliptic problems in non smooth domains, Pitman, London (1985). · Zbl 0695.35060 [5] Grisvard, P.) .- Problèmes aux limites dans des domaines avec points de rebroussement, Ann. Fac. Sc. ToulouseII, n° 3 (1995), pp. 561-578. · Zbl 0864.35018 [6] Ibuki, K.) .- Dirichlet problem for elliptic equations of second order in a singular domain of R2, J. Math. Kyoto Univ.14, n° 1 (1974), pp. 55-71. · Zbl 0281.35031 [7] Khelif, A.) . - Problèmes aux limites pour le laplacien dans un domaine à points cuspides, C. R. Acad. Sci.Paris287. Série A. (1983), pp. 1113-1116, et Thèse présentée à l’Université de Nice (juin 1978). · Zbl 0402.35015 [8] MAZ’YA, V.G.) et Plamenevskii, B.A.) .- Estimates in Lp and in Hölder classes and the Miranda-Agmon Maximum Principe for solutions of Elliptic Boundary Value Problems in domains with singular Points on the Boundary, Amer. Math. Soc. Transl.2 (1984), pp. 123-156; traduction de Math. Nachr.81 (1978), pp. 25-82. · Zbl 0554.35035 [9] Steux, J.-L.) . - Problème de Dirichlet pour le laplacien dans un domaine à point cuspide, C. R. Acad. Sci.Paris. 306, Série I (1988), pp. 773-776. · Zbl 0651.35016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.