Continuous dependence in hyperbolic problems with Wentzell boundary conditions

Giuseppe Maria Coclite; Angelo Favini; Gisèle Ruiz Goldstein; Jerome A. Goldstein; Silvia Romanelli

doi:10.3934/cpaa.2014.13.419

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January 2014, 13(1): 419-433. doi: 10.3934/cpaa.2014.13.419

Continuous dependence in hyperbolic problems with Wentzell boundary conditions

Giuseppe Maria Coclite ^1, , Angelo Favini ^2, , Gisèle Ruiz Goldstein ^3, , Jerome A. Goldstein ^4, and Silvia Romanelli ^1,

1.	Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari, Italy, Italy
2.	Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna
3.	The University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152, United States
4.	Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

Received January 2013 Revised May 2013 Published August 2013

Abstract
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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.

Keywords: Wentzell boundary conditions, higher order boundary operators., continuous dependence, Wave equation, semigroup approximation.

Mathematics Subject Classification: Primary: 35B30, 47D06; Secondary: 35L0.

Citation: Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419