Strong solvability of a unilateral boundary value problem for nonlinear parabolic operators (Q2460794)

This article deals with the strong solvability of the problem: \[ \begin{aligned} A(X,D2u)-u_t = f(X) &\quad\text{a.e. in } Q=\Omega\times\left]0, T\right[ \subset \mathbb{R}^{n+1}\\ u(x,0)=0 &\quad\text{a.e. in }\Omega\\ u\geq 0, \frac{\partial u}{\partial\nu}\leq 0, u\cdot\frac{\partial u}{\partial\nu} &\quad\text{a.e. in }\partial\Omega\times\left]0, T\right[, \end{aligned} \] where \(\Omega\) denotes a smooth, bounded and convex open set in \(\mathbb{R}^n\), \(\nu\) the outward normal derivative and \(X=(x,t)\) the point in the cylinder \(Q\). The operator \(A(X,\xi)\) is of Carathéodory type and satisfies a suitable ellipticity condition introduced by \textit{S. Campanato} [Ric. Mat. 40, Suppl., 129--140 (1991; Zbl 0796.35052)] that implies the ``nearness'' of the operator \(A(X,\xi)\) to the Laplace operator on the set of the functions satisying the boundary conditions on \(\Omega\times\left]0, T\right[\) (see conditions in (iii) of \textit{S. Campanato} [Ann. Mat. Pura Appl. (4) 167, 243--256 (1994; Zbl 0820.47050)]).