Solvability of the Dirichlet problem for a~general second-order elliptic equation

V. Zh. Dumanyan

Sbornik. Mathematics, Tome 202 (2011) no. 7, pp. 1001-1020

Résumé

The paper is concerned with studying the solvability of the Dirichlet problem for the second-order elliptic equation \begin{gather*} \begin{split} -\operatorname{div} (A(x)\nabla u)+(\overline b(x),\nabla u)-\operatorname{div} (\overline c(x)u)+d(x)u \\ \qquad=f(x)-\operatorname{div} F(x), \qquad x\in Q, \end{split} \\ u\big|_{\partial Q}=u_0, \end{gather*} in a bounded domain $Q\subset R_n$, $n\geqslant 2$, with $C^1$-smooth boundary and boundary condition $u_0\in L_2(\partial Q)$. Conditions for the existence of an $(n-1)$-dimensionally continuous solution are obtained, the resulting solvability condition is shown to be similar in form to the solvability condition in the conventional generalized setting (in $W_2^1(Q)$). In particular, the problem is shown to have an $(n-1)$-dimensionally continuous solution for all $u_0\in L_2(\partial Q)$ and all $f$ and $F$ from the appropriate function spaces, provided that the homogeneous problem (with zero boundary conditions and zero right-hand side) has no nonzero solutions in $W_2^1(Q)$. Bibliography: 14 titles.

Détail

Keywords: Dirichlet problem, solvability of the Dirichlet problem, second-order elliptic equation, $(n-1)$-dimensionally continuous solution.