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Regularity for elliptic equations with general growth conditions. (English) Zbl 0812.35042

The author studies elliptic equations in divergence form \[ D_ i (A^ i (x,u,Du))= b(x,u,Du), \tag \(*\) \] where the vector field \(A^ i (x,z,\xi)\) is Lipschitz with respect to all its variables. He assumes, among other conditions, that the matrix \((a^{ij})= \partial A/\partial\xi\) satisfies the conditions \(a^{ij} \zeta_ i \zeta_ j\geq g_ 1 (| \xi|)| \zeta|^ 2\), \(| a^{ij} |\leq g_ 2(| \xi|)\) for some positive functions \(g_ 1\) and \(g_ 2\) such that \[ g_ 2 (n^{1/2} t)=O \biggl( t^{-2} \biggl[\int_ 0^ t (g_ 1(s))^{1/2} ds\biggr]^ q \biggr) \tag \(**\) \] where \(q= 2n/ (n-2)\) if the dimension \(n>2\) and \(q\) is any positive number if \(n=2\). If \(g_ 2\) is some multiple of \(g_ 1\), the equation is uniformly elliptic and the regularity of the solution has been known for a long time. For the more general condition \((**)\), the regularity is new. The key step here is to prove a local gradient bound for a solution of the equation under hypothesis \((**)\). Such a result was proved by L. Simon [Indiana Univ. Math. J. 25, 821-855 (1976; Zbl 0346.35016)] under more general structure conditions than those given by Marcellini; however, the paper under review deals directly with weak solutions of \((*)\). In addition Marcellini provides a more extensive list of examples.

MSC:

35J60 Nonlinear elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0346.35016
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