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Dirichlet problem for a linear elliptic equation in unbounded domains with $L^2$-boundary data

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Dirichlet problem for a linear elliptic equation in unbounded domains with L2-boundary data RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA J. CHABROWSKI Dirichlet problem for a linear elliptic equation in unbounded domains with L2-boundary data Rendiconti del Seminario Matematico della Università di Padova, tome 71 (1984), p. 287-328. <http://www.numdam.org/item?id=RSMUP_1984__71__287_0> © Rendiconti del Seminario Matematico della Università di Padova, 1984, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Dirichlet Problem for a Linear Elliptic Equation in Unbounded Domains with L2 - Boundary Data. J. CHABROWSKI 1. Introduction. The main purposes of this paper are to investigate the Dirichlet problem for the elliptic equation in a half-space and a complement of a bounded open set. We shall refer to the second problem as the exterior Dirichlet problem. Given an open set Q c we denote by the Banach space of functions u in Z2(S~) having weak (distributional) derivatives Diu. (i = 1, ..., n) in L2(Q). A norm is introduced by defining The closure of in Wl,2(Q) is denoted by A local space consists of functions belonging to for every bounded open set ,~’ such that S~. To motivate our approach to the Dirichlet problem assume for simplicity that .~ is uniformly elliptic and the coefficients c (*) Indirizzo dell’A.: Department of Mathematics, University of Queens- land, St. Lucia Queensland 4067, Australia. 288 and f are measurable an

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