On a Nonlinear Elliptic Boundary Value Problem
نویسنده
چکیده
Consider a bounded domain G C R (_N>1) with smooth boundary T . Let L be a uniformly elliptic linear differential operator. Let y and ß be two maximal monotone mappings in R. We prove that, when y ? 2 satisfies a certain growth condition, given f £ L (G ) there is u € H (G) such that Lu + y(u) 3 f a.e. on G, and -du/d v e ß(u\ ) a.e. on T, where du/civ is the conormal derivative associated with L. 1. Let GC R (N > l) be abounded domain with smooth boundary Y. Consider the uniformly elliptic linear operator L22 = -D.U..(x)D.T2) + b.(x)D.u + c(x)u, a.. = a.. £ Cl(G); b., c e L°°(G) (i, j = 1, 2, • •, , N), ««(*)£ff> c|£|2, c> 0 constant, Vx £ G, f £ RN. (All functions and scalars that we consider are real.) Let y : R —> 2 be a maximal monotone mapping. The domain D(y) of y is the set of all numbers s such that y (s) / 0For each s £ D(y), y(s) is a closed interval and thus contains a unique element, which we denote by y (s), having smallest absolute value. We assume that the mapping y satisfies the condition (1) |y°(s)| > <p{s)\s\, Vs£D(y) with lim 0(s) = °°. |s| —«oo It can be verified that y induces a maximal monotone mapping y : L (G) —» 2 ( in a natural way: y(u) ={v £ L2(G)|iXx) £ yiu(x)) a.e.} (u £ L2(G)). Received by the editors February 8, 1974. AUS (UOS) subject classifications (1970). Primary 35J20, 35J60, 35J65; Secondary 47H05, 47H10, 47H15
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