HEAT FLOWS FOR A NONCONVEX SIGNORINI TYPE PROBLEM IN R~N

摘要

We prove the existence of a global heat flow u : Ω × R~+ → R~N, N > 1, satisfying a Signorini type boundary condition u(dΩ x R~+) C K, where Ω is a bounded domain in R~n, n≥ 2, and K is a nonconvex (not necessarily compact) set in R~n with boundary dK of class C~2. The function u(.,t) maps any smooth function on 0, such that as t —> oo, where u° is an extremal of the variational problem for the energy functions with the boundary obstacle <% and u°(dQ.) C J^. We show that u(x,t) is a weak solution to the nonstationary Signorini problem and obtain an estimate for the admissible singular set of u. A similar result is valid if an obstacle in M.N is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.