Estimation and Regularity for the Maximum Lower Solution of a Unilateral Problem

Abstract

Let Ω be a bounded open set of ℝ² with smooth boundary δΩ; T is a positive real number, Q denote the open cylinder Q = Ω × ]0, T[ and Σ = δΩ × ]0, T[, T[. ψ is a given function. We look at the following problem find a function u such that:

$$\left. {\begin{array}{*{20}{c}} \hfill {\begin{array}{*{20}{c}} \hfill {Eu = \frac{{\partial u}}{{\partial t}} - \Delta u \leqslant 0} \\ \hfill {u \leqslant \psi } \\ \end{array} } \\ \hfill {\begin{array}{*{20}{c}} {Eu(u - \psi ) = 0} \\ {u(x,0) = 0} \\ {{}^{u}\left| \sum \right. = 0} \\ \end{array} } \\ \end{array} } \right\}$$

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