On the regularity of linear elliptic and parabolic boundary value problems with nonsmooth data (Q2715830)

The paper is concerned with the regularity theory of boundary value problems (BVP) for linear second-order elliptic and parabolic equations. The weak form of the BVP is \(\langle L_Gu,w\rangle_G= \langle F,w\rangle_G\), where the coefficients of the operator \(L_G\) are bounded and measurable, the boundary value problem is of a mixed type and the boundary of the domain \(G \subset\mathbb{R}^n\) is not smooth, the functional \(F\) belongs to the dual space \(Y^\omega\) realized as an image of the space \(W_0^\omega\) by means of the operator \(L_G\). The space \(W_0^\omega\) is the intersection of the space of weak solutions of BVP with the Sobolev-Campanato space \(W^\omega= \{u\in L^2(G)\); \(D_\alpha u\in {\mathcal L}^{2,\omega}(G)\), \(\alpha=1, \dots,n\}\), \(0\leq\omega <n+2\). It is proved that, for any regular set \(G\) and for the elliptic operators \(L_G\), there is a number \(\overline \omega>n-2\) such that, for all \(\omega\), \(0\leq\omega <\overline\omega\), the elliptic operator \(L_G\) is an isomorphism from \(W_0^\omega(G)\) onto \(Y^\omega (G)\). If the exponent \(\omega\) is large enough, then the Hölder continuity of \(u\) is guaranteed. In the parabolic case, the author obtained regularity results of a similar character.