On solvability of variational Dirichlet problem for a class of degenerate elliptic operators

S. A. Iskhokov ; I. A. Yakushev

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 164-182

Résumé

The paper is devoted to investigation of unique solvability of the Dirichlet variational problem associated with integro-differential sesquilinear form \begin{equation*}\tag{*} B[u,\,v]=\sum_{j\in J}B_j[u,\,v], \end{equation*} where \begin{equation*} B_j[u,\,v]= \sum_{|k|=|l|=j}\int_{\Omega}\rho(x)^{2\tau_j}b_{kl}(x)u^{(k)}(x)\,\overline{v^{(l)}(x)}dx, \end{equation*} $\Omega$ — a bounded domain in the euclidian space $R^n$ with a closed $(n-1)$-dimensional boundary $\partial \Omega$, $\rho(x),\,x\in \Omega,$ — a regularized distance from a point $x\in\Omega$ to $\partial \Omega$, $k$ — a multi-index, $u^{(k)}(x)$ — a generalized derivative of multi-index $k$ of a function $u(x),\,x\in \Omega$, $b_{kl}(x)$ — bounded in $\Omega$ complex-valued functions, $J\subset \{1,\,2,\,\ldots,\,r\}$ and $\tau_j,\,j\in J,$ — real numbers. It is assumed that $r\in J$. A degeneracy of coefficients of the differential operator associated with the form (*), is said to be coordinated if there exist a number $\alpha$ such that $\tau_j=\alpha+j-r$ for all $j\in J.$ Otherwise it is called uncoordinated. The variational Dirichlet problem associated with the form (*) in the case of coordinated degeneracy of coefficients is well studied in many papers, where it is also assumed that the form (*) satisfies a coercivness condition. It should be mentioned that the case of uncoordinated degeneracy of the coefficients is fraught with some technical complexities and it was only considered in some separate papers. In this case with the aid of embedding theorems for spaces of differentiable functions with power weights leading forms $B_j[u,\,v],\, j\in J_2\subset J,$ are separated and it is proved that solvability of the variational Dirichlet problem is generally depends on the leading forms. We consider the case of uncoordinated degeneracy of coefficients of the operator under investigation and, in contrast to previously published works on this direction, it is allowed that the main form (*) does not obey coerciveness condition.

Détail

Keywords: variational Dirichlet problem, elliptic operator, uncoordinated degeneration, noncoercive form.