American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Geometric conditions for the existence of a rolling without twisting or slipping
  • CPAA Home
  • This Issue
  • Next Article
    Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations
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

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 & 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.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space,, Comm. Pure Appl. Math., 16 (1963), 121.  doi: 10.1002/cpa.3160160204.  Google Scholar

[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, (2009), 279.   Google Scholar

[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.  doi: 10.1007/s00233-008-9068-2.  Google Scholar

[5]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Graduate Texts in Mathematics, (2000).   Google Scholar

[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.  doi: 10.1007/s00028-002-8077-y.  Google Scholar

[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.  doi: 10.1002/mana.200910086.  Google Scholar

[8]

J. A. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford University Press, (1985).  doi: 10.1016/0022-1236(69)90020-2.  Google Scholar

[9]

J. A. Goldstein, Time dependent hyperbolic equations,, J. Functional Analysis, 4 (1969), 31.   Google Scholar

[10]

J. A. Goldstein and G. Reyes, Asymptotic equipartition of operator-weighted energies in damped wave equations,, {Asymptotic Analysis}, ().   Google Scholar

[11]

T. Kato, "Perturbation Theory for Linear Operators,", Die Grundlehren der mathematischen Wissenschaften, (1966).   Google Scholar

[12]

P. D. Lax, "Functional Analysis,", Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], (2002).   Google Scholar

[13]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I,", Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar

[14]

H. Triebel, "Theory of Function Spaces,", Monographs in Mathematics, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

show all references

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.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space,, Comm. Pure Appl. Math., 16 (1963), 121.  doi: 10.1002/cpa.3160160204.  Google Scholar

[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, (2009), 279.   Google Scholar

[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.  doi: 10.1007/s00233-008-9068-2.  Google Scholar

[5]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Graduate Texts in Mathematics, (2000).   Google Scholar

[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.  doi: 10.1007/s00028-002-8077-y.  Google Scholar

[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.  doi: 10.1002/mana.200910086.  Google Scholar

[8]

J. A. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford University Press, (1985).  doi: 10.1016/0022-1236(69)90020-2.  Google Scholar

[9]

J. A. Goldstein, Time dependent hyperbolic equations,, J. Functional Analysis, 4 (1969), 31.   Google Scholar

[10]

J. A. Goldstein and G. Reyes, Asymptotic equipartition of operator-weighted energies in damped wave equations,, {Asymptotic Analysis}, ().   Google Scholar

[11]

T. Kato, "Perturbation Theory for Linear Operators,", Die Grundlehren der mathematischen Wissenschaften, (1966).   Google Scholar

[12]

P. D. Lax, "Functional Analysis,", Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], (2002).   Google Scholar

[13]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I,", Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar

[14]

H. Triebel, "Theory of Function Spaces,", Monographs in Mathematics, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[1]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
[2]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Enrico Obrecht, Silvia Romanelli. Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2475-2487. doi: 10.3934/cpaa.2016045
[3]

Maike Schulte, Anton Arnold. Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme. Kinetic & Related Models, 2008, 1 (1) : 101-125. doi: 10.3934/krm.2008.1.101
[4]

Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020239
[5]

Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401
[6]

Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041
[7]

Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034
[8]

Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102
[9]

Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259
[10]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[11]

Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
[12]

Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[13]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441
[14]

Michael Renardy. A backward uniqueness result for the wave equation with absorbing boundary conditions. Evolution Equations & Control Theory, 2015, 4 (3) : 347-353. doi: 10.3934/eect.2015.4.347
[15]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367
[16]

Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603
[17]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001
[18]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[19]

Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
[20]

Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences