Approximation durch Lösungen partieller Differentialgleichungen (Q1068280)

Let \(\Omega \subset {\mathbb{R}}^ m\) (m\(\geq 2)\) be a bounded domain, \[ Lu(x)=\sum^{m}_{i,k=1}a_{ik}(x)\frac{\partial^ 2u}{\partial x_ i\partial x_ k}+\sum^{m}_{i=1}b_ i(x)\frac{\partial u}{\partial x_ i}+c(x)u(x);\quad x=(x_ 1,...,x_ m) \] is an elliptic differential operator, \(a_{ik}\in C^ 2({\bar \Omega})\) \(b_ i\in C^ 1({\bar \Omega})\), \(c\in C({\bar \Omega})\). \(\Gamma \in C^ 1\) is a closed (m-1)-dimensional surface in the interior of \({\bar \Omega}\) and \(\Omega_ i\) the interior part of \(\Gamma\) in \(\Omega\) \((\partial \Omega_ i=\Gamma)\). Suppose, the Dirichlet problem \(Lu=0\) in \(\Omega\), \(u|_{\partial \Omega}=0\) has n (\(\geq 0)\) linearly independent eigensolutions \(u^{(1)},...,u^{(n)}\). Suppose further that the problem \(Lu=0\) in \(\Omega_ i\) and \(u|_{\Gamma}=0\) has p eigensolutions, that of the adjoint problem are denoted by \(\tilde v^{(1)},...,\tilde v^{(p)}\) (which are chosen orthonormal in \(L_ 2(\Omega_ i))\). Consider the following linear and bounded functionals on C(\(\Gamma)\): \[ l_ k(h)=\int_{\Gamma}h(x)(d\tilde v^{(k)}(x)/dn_ x) d\sigma (x)\quad k=1,2,...,p \] where \(n_ x\) is the conormal at x to \(\Gamma\), \(\sigma\) denotes the Lebesgue surface- measure, \[ a(x)=[\sum^{m}_{i=1}(\sum^{m}_{k=1}a_{ik}(x)\cos (n,x_ k))^ 2]^{1/2}. \] The following notation is introduced: \(H(\Gamma)=\cap^{p}_{k=1}\ker l_ k\) if \(p>0\); \(H(\Gamma)=C(\Gamma)\) if \(p=0\) and \(M(\Gamma)=\{u|_{\Gamma}:\) \(L(u)=0\) in \(\Omega\), \(u|_{\partial \Omega}=f\}\), where \(f\in C(\partial \Omega)\) and \(f=0\) on \(\partial \Omega \setminus V\), V is an open subdomain of \(\partial \Omega\). The main result of the paper states that the restrictions to \(\Gamma\) of the solutions of the Dirichlet problem in M(\(\Gamma)\) are dense in the space H(\(\Gamma)\).