Time and space Sobolev regularity of solutions to homogeneous parabolic equations
This paper concerns initial-boundary value problems of the type \(\partial_t u= Eu\) in \((0,T]\times \Omega\), \(u= 0\) on \((0,T)\times \partial\Omega\), \(u= u_0\) on \(\{0\}\times \Omega\) with a second-order uniformly elliptic operator in divergence form \[ Eu:= \sum^n_{i,j=1} \partial_i(a_{ij}(x)) \partial_ju)+ \sum^n_{i=1} \partial_i(b_i(x)u)+ c(x)u. \] The author formulates necessary and sufficient conditions on the initial function \(u_0\) such that the solution has Sobolev regularity \[ \partial^k_t u\in W^{\alpha,p}(0, T; L^p(\Omega))\cap L^p(0, T; H^{2\alpha,p}_*(\Omega)) \] for \(k= 0,1,\dots\), \(0< \alpha\leq 1\), \(0\leq p<\infty\). Here \(W^{\alpha,p}(0, T; L^p(\Omega))\) is the space of all \(u\in L^p((0,T)\times \Omega)\) such that \[ \int^T_0 \int_\Omega\|u(t,\cdot)- u(s,\cdot)\|_{L^p(\Omega)}|t-s|^{-1-\alpha p}ds dt< \infty, \] and \(H^{2,\alpha,p}_*(\Omega)\) is the real interpolation space \((L^p(\Omega), D(E))_{\alpha, p}\), where \(D(E):= \{u\in H^{1,p}_0(\Omega): Eu\in L^p(\Omega)\}\). Unfortunately, the regularity, needed for the boundary \(\partial\Omega\) and for the coefficients \(a_{ij}\), \(b_i\) and \(c\) is not formulated. In the paper there are no proofs in fact, but only citations, so these conditions cannot be reconstructed by the proofs.
second-order uniformly elliptic operator in divergence form
necessary and sufficient conditions on the initial function
scientific article; zbMATH DE number 1281954