On the dimension of the~solution space of elliptic systems in unbounded domains

A. A. Kon'kov

Sbornik. Mathematics, Tome 80 (1995) no. 2, pp. 411-434

Résumé

This article is a study of the Dirichlet problem $$ \begin{cases} Lu=0\text{in}\ \Omega, \\ \partial^\alpha u\big|_{\partial \Omega}=0,|\alpha|\leqslant m-1, \end{cases} $$ where $\Omega\subset R^n$ is an open (possibly unbounded) set, $\alpha=(\alpha_1,\dots,\alpha_n)$ is a multi-index, $|\alpha|=\alpha_1+\dots+\alpha_n$, $$ L=\sum_{|\alpha|=|\beta|=m}\partial^\alpha \bigl(a_{\alpha\beta}(x)\partial^\beta\bigr), $$ and the coefficients $a_{\alpha\beta}(x)$ are $N\times N$ matrices.