Selfadjoint extensions of the Dirichlet problem operator in weighted function spaces

S. A. Nazarov

Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 229-247

Résumé

The operators considered here are symmetric in $L_2$ with the weight $|x|^{2\sigma}$ and correspond to Dirichlet problems for formally selfadjoint elliptic (in the Petrovskii sense) systems of differential equations of order $2m$ in a bounded domain $\Omega\subset\mathbf R^n$, $O\in\Omega$. All selfadjoint extensions of the operators are listed for every $\sigma\geqslant-m$ with exception of a countable set of half-integer exponents. It is shown that with increasing $\sigma$ the asymptotic conditions at $O$ corresponding to these extensions include all the higher derivatives of the fundamental solution. Analogous assertions concerning the case $O\in\partial\Omega$ are given. Bibliography: 20 titles.