Continuous dependence in hyperbolic problems with Wentzell boundary conditions
Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli · Communications on Pure & Applied Analysis · 2013
Let $\Omega$ be a smooth bounded domain in $R^N$ and let\begin{eqnarray}Lu=\sum_{j,k=1}^N \partial_{x_j}\left(a_{jk}(x)\partial_{x_k} u\right),\end{eqnarray}in $\Omega$ and\begin{eqnarray}Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\partial_{\tau_k}\left(b_{jk}(x)\partial_{\tau_j}u\right)=0,\end{eqnarray}on $\partial\Omega$define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary.Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space.If we have a sequence of such operators $S_0,S_1,S_2,...$ with corresponding coefficients\begin{eqnarray}\Phi_n=(a_{jk}^{(n)},b_{jk}^{(n)}, \beta_n,\gamma_n,q_n)\end{eqnarray}satisfying $\Phi_n\to\Phi_0$ uniformly as $n\to\infty$, then $u_n(t)\to u_0(t)$ where $u_n$ satisfies\begin{eqnarray}i\frac{du_n}{dt}=S_n^m u_n,\end{eqnarray}or\begin{eqnarray}\frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0,\end{eqnarray}or\begin{eqnarray}\frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0,\end{eqnarray}for $m=1,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$.This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.