Boundary value problems with nonuniformly characteristic boundary
Consider \(\Omega\subset \mathbb{R}^d\) and \(L\equiv \sum_{1\leq j\leq d} A_j(x)\partial_j+ B(x)\) with Lipschitzian coefficients. \(A_n(x)\equiv \sum n_j A_j\) denotes the boundary matrix. Suppose that there is a \(C^1\)-embedded \(d-2\)-dimensional submanifold \(\Gamma\) of \(\partial\Omega\) such that \(\text{rank}(A_n)\) is constant on each component of the complement of \(\Gamma\) in \(\partial\Omega\). Homogeneous boundary conditions \(u(x)\in N(x)\) for \(x\in \partial\Omega\) are imposed, where for each \(x\in \partial\Omega\), \(N(x)\) is a linear subspace of \(\mathbb{C}^k\). On each component of the complement of \(\Gamma\) in \(\partial \Omega\), \(N(x)\) is \(C^1\). Hypothesis: For each \(\underline x\in \Gamma\) there is a relatively open subset \({\mathcal O}\subset \partial\Omega\), \(x\in {\mathcal O}\) such that: 1. \({\mathcal O}\) is \(C^1\) diffeomorphic to a ball \(\{|y|< 1\}\) with \(\Gamma\) going to the equatorial plane \(\{y_1= 0\}\). 2. There are two \(C^1\) vector subbundles \(N_{\text{big}}(x)\) and \(N_{\text{small}}(x)\) of \({\mathcal O}\times \mathbb{C}^k\) such that i. \(N_{\text{big}}(x)\supset \ker A_n(x)\) for all \(x\in {\mathcal O}\), ii. \(N_{\text{big}}(x)\supset N_{\text{small}}(x)\) for all \(x\in {\mathcal O}\), iii. \(N= N_{\text{big}}\) on one side of \({\mathcal O}\backslash \Gamma\) and \(N= N_{\text{small}}\) on the other. Definition: For \(f\in L^2(\Omega)\), a function \(u\in L^2(\Omega)\) is a weak solution of \(Lu= f\), \(u\in N\) at \(\partial\Omega\) iff \(Lu= f\) in the sense of distributions on \(\Omega\) and \(u\in N\) on each component of \(\partial\Omega \backslash\Gamma\). Theorem: If \(f\in L^2(\Omega)\), and \(u\in L^2(\Omega)\) is a weak solution of \(Lu= f\), and \(u\in N\), then \(u\) is a strong solution, that is there exist \((u_m)\subset C^\infty(\overline\Omega)\) with \(u_m(x)\in N\) for all \(x\in \partial\Omega \backslash\Gamma\) and \(u_m\to u\) and \(Lu_m\to f\) in \(L^2(\Omega)\).
scientific article; zbMATH DE number 811446