\(L_{p}\)-solvability of the Dirichlet problem for second order parabolic equations
This paper concerns weak solutions to divergence form equations of the type \[ \partial_t u+ \sum^n_{j,k=1}\partial_j a_{jk}(x, t)\partial_k u= \sum^n_{j=1} \partial_j f_j(x,t)\quad\text{in }\Omega_T:= \Omega\times [0,T] \] with homogeneous initial and Dirichlet boundary conditions. Here \(\Omega\subset\mathbb{R}^n\) is a bounded domain, the coefficient functions \(a_{jk}:\overline\Omega\times [0,T]\to \mathbb{R}\) are continuous and satisfy a uniform ellipticity condition, and \(f\in L^p(Q_T; \mathbb{R}^n)\). There are formulated regularity conditions on \(\partial\Omega\) (in terms of a so-called harmonic measure on \(\partial\Omega)\) which are necessary and sufficient for unique solvability and for the a priori estimates \[ \| u\|_{L^p(Q_T)}+ \|\nabla_x u\|_{L^p(Q_T;\mathbb{R}^n)}\leq \text{const}\| f\|_{L^p(Q_T; \mathbb{R}^n)} \] for all \(p\geq 2\).
scientific article; zbMATH DE number 5924724
scientific article