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

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.