Sulle singolarità eliminabili delle soluzioni deboli di equazioni ellittiche quasi lineari di ordine 2m
Given a bounded domain \(\Omega \subset {\mathbb{R}}^ n\) and a compact \(S\subset \Omega\) of measure zero, consider the quasilinear elliptic problem \[ (*)\quad \sum_{| \alpha | \leq m}\int_{\Omega}F_{\alpha}(x,u,D^ mu)D^{\alpha}\phi dx=0\quad(\phi \in C_ 0\!^{\infty}(\Omega \backslash S)). \] The problem of removable singularities of (weak) solutions to (*) can be formulated as follows: Given a suitable subclass \(A\subset W^{m,p}\!\!\!_{loc}(\Omega \backslash S),\) find conditions on S which guarantee that each \(u\in A\) is a solution to (*) (with \(\phi \in C_ 0\!^{\infty}(\Omega))\) of class \(W^{m,q}\!\!\!_{loc}(\Omega)\) for some \(q>1\). A result in this spirit is due to \textit{V. G. Maz'ja} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 27, 116-130 (1972; Zbl 0351.35035)] in the case \(A=W_{loc}\!\!\!^{m,p}(\Omega \backslash S)\cap L^{\infty}(\Omega).\) In the present paper the authors give some parallel results for the cases \(A_ 1=W^{m,p}\!\!\!_{loc}(\Omega \backslash S)\cap L^ p\!_ s(\Omega \backslash S), A_ 2=W^{m,p}\!\!\!_{loc}(\Omega \backslash S)\cap L^ q(\Omega \backslash S)\) and \(A_ 3=W^{m,r}(\Omega \backslash S).\) The corresponding conditions are formulated in terms of the so-called ''d- dimension'' of a set M, introduced by \textit{M. Marcus} [Amer. J. Math. 90, 197-213 (1968; Zbl 0153.424)].
scientific article; zbMATH DE number 3849691