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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

 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

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.
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
##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. [2] S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., 16 (1963), 121-239. doi: 10.1002/cpa.3160160204. [3] G. M. Coclite, A Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis, In "Advances in Nonlinear Analysis: Theory, Methods and Applications" (S. Sivasundaran ed.), vol. 3, pages 279-292, Cambridge Scientific Publishers Ltd., Cambridge, 2009. [4] G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary conditions for the Wentzell Laplacian, Semigroup Forum, 77 (2008), 101-108. doi: 10.1007/s00233-008-9068-2. [5] K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. [6] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary conditions, J. Evol. Equ., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. [7] A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086. [8] J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, Oxford, 1985. doi: 10.1016/0022-1236(69)90020-2. [9] J. A. Goldstein, Time dependent hyperbolic equations, J. Functional Analysis, 4 (1969), 31-49. [10] J. A. Goldstein and G. Reyes, Asymptotic equipartition of operator-weighted energies in damped wave equations,, {Asymptotic Analysis}, (). [11] T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, vol. 132, Springer-Verlag, New York, 1966. [12] P. D. Lax, "Functional Analysis," Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. [13] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I," Die Grundlehren der mathematischen Wissenschaften, vol. 181, Springer-Verlag, New York, 1972. [14] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

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##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. [2] S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., 16 (1963), 121-239. doi: 10.1002/cpa.3160160204. [3] G. M. Coclite, A Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis, In "Advances in Nonlinear Analysis: Theory, Methods and Applications" (S. Sivasundaran ed.), vol. 3, pages 279-292, Cambridge Scientific Publishers Ltd., Cambridge, 2009. [4] G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary conditions for the Wentzell Laplacian, Semigroup Forum, 77 (2008), 101-108. doi: 10.1007/s00233-008-9068-2. [5] K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. [6] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary conditions, J. Evol. Equ., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. [7] A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086. [8] J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, Oxford, 1985. doi: 10.1016/0022-1236(69)90020-2. [9] J. A. Goldstein, Time dependent hyperbolic equations, J. Functional Analysis, 4 (1969), 31-49. [10] J. A. Goldstein and G. Reyes, Asymptotic equipartition of operator-weighted energies in damped wave equations,, {Asymptotic Analysis}, (). [11] T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, vol. 132, Springer-Verlag, New York, 1966. [12] P. D. Lax, "Functional Analysis," Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. [13] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I," Die Grundlehren der mathematischen Wissenschaften, vol. 181, Springer-Verlag, New York, 1972. [14] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
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