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We prove the unique solvability for the Poisson and heat equations innon-smooth domains $\Omega\subset \mathbb{R}^d$ in weighted Sobolev spaces. Thezero Dirichlet boundary condition is considered, and domains are merely assumedto admit the Hardy inequality: $$\int_{\Omega}\Big|\frac{f(x)}{d(x,\partial\Omega)}\Big|^2\,\,\mathrm{d} x\leqN\int_{\Omega}|\nabla f|^2 \,\mathrm{d} x\,\,\,\,,\,\,\,\, \forall f\inC_c^{\infty}(\Omega)\,. $$ To describe the boundary behavior of solutions, weintroduce a weight system that consists of superharmonic functions and thedistance function to the boundary. The results provide separate applicationsfor the following domains: convex domains, domains with exterior conecondition, totally vanishing exterior Reifenberg domains, conic domains, anddomains $\Omega\subset\mathbb{R}^d$ which the Aikawa dimension of $\Omega^c$ isless than $d-2$.

Sobolev space theory for Poisson's and the heat equations in non-smooth domains via superharmonic functions and Hardy's inequality