We obtain an approximation result of the weak solutions to elliptic and parabolic equations with Dirichlet boundary conditions. We show that the weak solution can be obtained with a limit of approximations by regularizing the nonlinearities and approximating the domains.

We obtain a boundary pointwise gradient estimate on a parabolic half cube $Q_{2R}\cap \{(x^1,x^{\prime },t)\in {\mathbb{R}}^{n+1}:x^1>0\}$ for nonlinear parabolic equations with measurable nonlinearities, which are only assumed to be measurable in $x^1$-variable. The estimates are obtained in terms of Riesz potential of the right-hand side measure and the oscillation of the boundary data, where the boundary data is given on $Q_{2R}\cap \{(x^1,x^{\prime },t)\in {\mathbb{R}}^{n+1}:x^1=0\}$.