Strongly nonlinear parabolic variational inequalities (Q5906517)

The paper extend some previous result of Brézis, Browder and Landes to a class of nonlinear variational inequalities. More precisely consider the problem: Find \(u\in K\cap L^ 2(Q)\cap L^ 1(0,T,W^{1,1}_ 0(\Omega))\) such that \[ (\partial_ t u, v-u)+ (A(u),v- u)\geq (f,v- u)\quad \forall v\in C^ 1(0,T,C^ \infty_ 0(\Omega))\cap K,\;u(0)= 0,\tag{1} \] where \(\Omega\) is a bounded, regular open set in \(\mathbb{R}^ n\), \(Q= \Omega\times (0,T)\), \(K\) is a closed convex subset of \(C(0,T,L^ 2(\Omega))\), \(0\in K\) and \[ A(u)(x,t)= \sum^ n_{i=1} \partial_{x_ i}(A_ i(x,t,\partial_{x_ i} u))+ A_ 0(x,t,u(x,t)), \] where \(A_ i(x,t,r)\) is continuous in \(t\) and \(r\) and measurable in \(x\), \(A_ i(x,t,0)= 0\), \(\sup_{| r|\leq s} | A_ i(x,t,r)|\) \(\leq h_ s(x,t)\), \(h_ s\in L^ 1(Q)\) and satisfies some monotonicity and coerciveness assumptions on \(r\). By using the Galerkin method an existence theorem is established for problem (1).