Quasilinear stochastic elliptic equations with reflection (Q1890714)

The authors consider the following nonlinear stochastic elliptic equation with Dirichlet boundary condition on a bounded domain \(D\) of \(\mathbb{R}^ k\), \(k = 1,2,3\), \[ \begin{cases} -\Delta u(x) + f \bigl( x,u(x) \bigr) = \dot w(x) + \eta,\\ u |_{\delta D} = 0, \end{cases} \tag{1} \] where \(\{\dot w(x), x \in D\}\) is a white noise on \(D\) and \(f\) a measurable function from \(D \times \mathbb{R}\) into \(\mathbb{R}\). A solution of (1) is a pair \((u, \eta)\) such that \(u = (u(x), x \in \overline D)\) is a nonnegative, continuous process on \(\overline D\) and \(\eta (dx)\) is a random measure on \(D\) satisfying \(\int_ D ud \eta = 0\). This condition forces the process \(u\) to be nonnegative. It is proved that if \(f\) is locally bounded, continuous and nondecreasing as a function of the second variable, then there exists a unique solution \((u, \eta)\) of equation (1). After transforming this problem into a deterministic one, the authors construct a solution by means of the penalization method. The uniqueness is obtained by a classical method in elliptic variational inequalities.