Quasilinear Elliptic Systems in Divergence Form with Weak Monotonicity
Norbert Hungerbühler · 1999
. We consider the Dirichlet problem for the quasilinear elliptic system \\Gamma div oe(x; u(x); Du(x)) = f on\\Omega u(x) = 0 on @\\Omega for a function u :\\Omega ! R m , where\\Omega is a bounded open domain in R n . For arbitrary right hand side f 2 W \\Gamma1;p 0 we prove existence of a weak solution under classical regularity, growth and coercivity conditions, but with only very mild monotonicity assumptions. Contents 1. Introduction 83 2. Galerkin approximation 85 3. The Young measure generated by the Galerkin approximation 86 4. Passage to the limit 86 References 90 1. Introduction On a bounded open domain\\Omega ae R n we consider the Dirichlet problem for the quasilinear elliptic system \\Gamma div oe(x; u(x); Du(x)) = f on\\Omega (1) u(x) = 0 on @\\Omega (2) for a function u :\\Omega ! R m . Here, f 2 W \\Gamma1;p 0(\\Omega\\Gamma for some p 2 (1; 1), and oe satisfies the conditions (H0)--(H2) below. A feature of this paper is that we treat a class of probl...