On the Dirichlet problem for a second-order elliptic equation

A. K. Gushchin

Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 19-66

Résumé

The function space $C_{n-1}(\overline Q)$, $C(\overline Q)\subset C_{n-1}(\overline Q)\subset L_2(Q)$, where $Q$ is a bounded domain in $\mathbf R_n$, consists of elements that on sets of positive $(n-1)$-dimensional Hausdorff measure have traces with a property analogous to joint continuity. For $\partial Q\in C^1$ the set of traces of the functions in $C_{n-1}(\overline Q)$ on $\partial Q$ coincides with $L_2(\partial Q)$, and the imbedding $W_2^1(Q)\subset C_{n-1}(\overline Q)$ is valid. Solutions of the Dirichlet problem in $C_{n-1}(\overline Q)$ are considered for the elliptic equation $$ \sum_{i,j=1}^n(a_{ij}(x)u_{x_i})_{x_j}=f,\quad x\in Q;\qquad u|_{\partial Q}=u_0. $$ Under the assumption that the normal to $\partial Q$ and the coefficients of the equation satisfy the Dini condition on $\partial Q$, it is established that for all $u_0\in L_2(\partial Q)$ and $f\in W_2^{-1}(Q)$ there is a unique solution that depends continuously on $u_0$ and $f$. It is proved that in this situation the solution in $C_{n-1}(\overline Q)$ coincides with the concept of a solution in $W^1_{2,\mathrm{loc}}$ introduced by Mikhailov. Bibliography: 39 titles.