A Simple Approach to a Dirichlet Problem Related to the Kardar-Parisi-Zhang Equation with Singular Coefficients

We look for solutions \(u \in W_0^{1,2}(\Omega )\) $u\in W_0^{1,2}(\mathrm{\Omega })$ of the KPZ-equation \(\begin{aligned} - \textrm{div}(M(x)\nabla u ) +a(x)\,u = \sqrt{\;[b(x)]^2+ |E\cdot \nabla u|^2 \;}+f (x), \end{aligned}\) $$\begin{array}{c}\hfill -\text{div}(M(x)\mathrm{\nabla }u)+a(x)u=\sqrt{{[b(x)]}^2+{|E·\mathrm{\nabla }u|}^2}+f(x),\end{array}$$ with zero boundary condition.