The Perron solution for elliptic equations without the maximum principle
In this work on classical potential theory the Dirichlet boundary value problem is considered for the solutions \(u:\Omega\to\mathbb{C}\) of a linear second order equation \(\mathcal{A}(u)=0\) in a bounded domain \(\Omega\) in \(\mathbb{R}^d\). The emphasis is on domains with irregular boundaries.\N\NDefine\N\[\N\langle\mathcal{A}(u), \psi\rangle = \sum_{k,l=1}^2 \int_\Omega a_{kl}(\partial_l u)\overline{\partial_k\psi} dx + \sum_{k=1}^2 \int_\Omega b_k u(\overline{\partial_k\psi}) dx + \sum_{k=1}^2 \int_\Omega c_k (\partial_k u) \overline\psi dx + \int c u \overline\psi dx \N\]\Nwhen \(u\in H_{loc}^1(\Omega)\) and \(\psi\in C_0^\infty (\Omega)\). The function \(u\in H_{loc}^1(\Omega)\) is called \(\mathcal{A}\)-harmonic if \(\mathcal{A}(u)=0\), i. e., \(\langle\mathcal{A}(u), \psi\rangle=0\) for all test functions \(\psi\in C_0^\infty (\Omega)\). We may assume that \(u\in C(\Omega)\), but not that \(u\in C(\overline\Omega)\).\N\NThe coefficients \(a_{kl}:\Omega\to\mathbb{R}\), \(c_k:\Omega\to\mathbb{R}\), \(b_k:\Omega\to\mathbb{C}\) and \(c: \Omega\to\mathbb{C}\) are bounded measurable functions. The assumptions\N\begin{itemize}\N\item \(\Re \sum a_{kl} \xi_k \overline\xi_l \geq |\xi|^2\)\N\item \(\mathcal{A}(u)=0\) for \(u\in H_0^1(\Omega)\) \(\Leftrightarrow\) \(u=0\)\N\end{itemize}\Nare made. No further assumptions are needed on the operator \(\mathcal{A}\).\N\NGiven continuous boundary values \(\phi\in C(\partial\Omega)\) the Dirichlet probem consists in constructing an \(\mathcal{A}\)-harmonic function \(u\in H_{loc}^1(\Omega)\) such that ``\(u|_{\partial\Omega}=\phi\)'' in some convenient sense.\NIf the boundary \(\partial\Omega\) is irregular, a solution \(u\in C(\overline\Omega)\) with \(u|_{\partial\Omega}=\phi\) cannot always exist, i. e., continuity up to the boundary fails.\NThe authors define a (unique) solution, which they call ``the Perron solution'', by abuse of notion. For the ordinary Laplace equation, it becomes the proper Perron solution. \NA remarkable feature is that the comparison principle is not generally valid (without further assumptions).\N\NSeveral equivalent definitions are provided for these ``Perron solutions''. One of them is to exhaust \(\Omega\) by regular subdomains. This approach was used by N. Wiener. The irregular boundary points and Wiener's Criterion are carefully analyzed. Kellogg's theorem is valid.
scientific article; zbMATH DE number 7932322
scientific article