Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

Abstract

We prove the existence of a global heat flow u : Ω × $$ {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}$$ , N > 1, satisfying a Signorini type boundary condition u(∂Ω × $$ {\mathbb{R}^{+}}$$ ) ⊂ , where Ω is a bounded domain in $$ {\mathbb{R}^{n}}$$ ), $$ n \geqslant 2 $$ , and is a nonconvex (not necessarily compact) set in $$ {\mathbb{R}^{N}}$$ ) with boundary ∂ of class C 2. The function u(·, t) maps any smooth function φ on $$ \bar{\Omega } $$ such that φ(∂Ω) ⊂ to u 0 as t → ∞, where u 0 is an extremal of the variational problem for the energy functions with the boundary obstacle and u 0(∂Ω) ⊂ . 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 $$ {\mathbb{R}^{N}}$$ is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.