Abstract:
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.
Keywords:Dirichlet problem, solvability of the Dirichlet problem, second-order elliptic equation, $(n-1)$-dimensionally continuous solution.
Fulltext:
PDF file (555 kB)
