April  2020, 19(4): 2147-2195. doi: 10.3934/cpaa.2020095

PDE problems with concentrating terms near the boundary

1. 

Grupo de Dinámica No Lineal, Universidad Pontificia Comillas de Madrid, C/ Alberto Aguilera 23, 28015 Madrid, Spain

2. 

Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid. 28040 Madrid, Spain

3. 

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM

*Corresponding author

Dedicated to Professor Tom´as Caraballo on occasion of his Sixtieth Birthday

Received  May 2019 Revised  September 2019 Published  January 2020

Fund Project: Partially supported by Project MTM2016-75465, MINECO, Spain and FIS2016-78883-C2-2-P(AEI/FEDER, U.E.). Partially supported by Severo Ochoa project SEV-2015-0554 (MINECO).

In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modification in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions.

Citation: Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. PDE problems with concentrating terms near the boundary. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2147-2195. doi: 10.3934/cpaa.2020095
References:
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R. Adams, Sobolev Spaces, Academic Press, Boston, 1978.

[2]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in, Schmeisser/Triebel: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.

[3]

G. S. Aragão and F. D. M. Bezerra, Upper semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary, J. Math. Anal. Appl., 462 (2018), 871-899.  doi: 10.1016/j.jmaa.2017.12.047.

[4]

G. S. Aragão and S. M. Bruschi, Concentrated terms and varying domains in elliptic equations: Lipschitz case, Math. Methods Appl. Sci., 39 (2016), 3450-3460.  doi: 10.1002/mma.3791.

[5]

G. S. AragãoA. L. Pereira and M. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differential Equations, 26 (2014), 871-888.  doi: 10.1007/s10884-014-9412-z.

[6]

G. S. Aragão and S. M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. Differential Equations, 253 (2012), 2573-2592.  doi: 10.1016/j.jde.2012.07.008.

[7]

G. S. AragãoA. L. Pereira and M. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Methods Appl. Sci., 35 (2012), 1110-1116.  doi: 10.1002/mma.2525.

[8]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations, 156 (1999), 376-406.  doi: 10.1006/jdeq.1998.3612.

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J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Attractors of parabolic problems with critical nonlinearities, uniform bounds, Comm. P. D. E.'s, 25 (2000), 1-37.  doi: 10.1080/03605300008821506.

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J. M. Arrieta and A. Jiménez-Casas, A. Rodríguez-Bernal, Nonhomogeneous flux condition as limit of concentrated reactions, Revista Iberoamericana de Matematicas, 24 (2008), 183–211.

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J. M. Arrieta, A. Nogueira and M. C. Pereira, Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary, Discrete and Continuous Dynamical Systems, to appear.

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J. M. ArrietaA. Rodríguez–Bernal and J. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary, Proceedings of The Royal Society of Edinburgh, 138A (2008), 223-237.  doi: 10.1017/S0308210506000813.

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J. M. Arrieta and E. Santamaría, Distance of attractors of reaction-diffusion equations in thin domains, Journal of Differential Equations, 263 (2017), 5459-5506.  doi: 10.1016/j.jde.2017.06.023.

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C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

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C. CavaterraC. GalM. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399.  doi: 10.1016/j.na.2009.11.002.

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G. A. ChechkinD. CioranescuA. Damlamian and A. L. Piatnitski, On boundary value problem with singular inhomogeneity concentrated on the boundary, J. Math. Pures Appl., 98 (2012), 115-138.  doi: 10.1016/j.matpur.2011.11.002.

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J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for monotone semigroups with applications to evolutionary equations, Journal of Differential Equations, 249 (2010), 485-525.  doi: 10.1016/j.jde.2010.04.006.

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P. CornilleauJ. P. Loheác and A. Osses, Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers, J. of Dynamical and Control Systems, 16 (2010), 163-188.  doi: 10.1007/s10883-010-9088-6.

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J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math. Z., 210 (1992), 413-439.  doi: 10.1007/BF02571805.

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J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[22]

J. Fernández Bonder, E. Lami Dozo and J. D. Rossi, Symmetry properties for the extremals of the Sobolev trace embedding, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 21 (2004), 795–805. doi: 10.1016/j.anihpc.2003.09.005.

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J. Fernández Bonder and J. D. Rossi, On the existence of extremals for the Sobolev trace embedding theorem with critical exponent, Bull. London Math. Soc., 37 (2005), 119-125.  doi: 10.1112/S0024609304003819.

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C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[25]

G. GilardiA. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.  doi: 10.3934/cpaa.2009.8.881.

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M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

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M. Grobbelaar-van Dalsen and N. Sauer, Solutions in Lebesgue spaces of the Navier-Stokes equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A., 123 (1993), 745-761.  doi: 10.1017/S0308210500030948.

[28]

Y. D. GolovatyD. GómezM. Lobo and E. Pérez, On vibrating membranes with very heavy thin inclusions, Math. Models Methods Appl. Sci., 14 (2004), 987-1034.  doi: 10.1142/S0218202504003520.

[29]

D. GómezM. LoboS. A. Nazarov and E. Pérez, Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues, J. Math. Pures Appl., 85 (2006), 598-632.  doi: 10.1016/j.matpur.2005.10.013.

[30]

J. K. Hale, Asymptotic Behavior of Dissipative System, 1988.

[31]

J. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326.  doi: 10.1007/BF01790353.

[32]

S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, J. of Differential Equations, 145 (1998), 184-215.  doi: 10.1006/jdeq.1997.3385.

[33]

A. Jiménez–Casas and A. Rodríguez-Bernal, Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary, Nonlinear Analysis T. M. A., 71 (2009), 2377-2383.  doi: 10.1016/j.na.2009.05.036.

[34]

A. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. and Appl., 379 (2011), 567-588.  doi: 10.1016/j.jmaa.2011.01.051.

[35]

A. Jiménez-Casas and A. Rodríguez-Bernal, Dynamic boundary conditions as a singular limit of parabolic problems with terms concentrating at the boundary, Dynamics of Partial Differential Equations, 9 (2012), 341-368.  doi: 10.4310/DPDE.2012.v9.n4.a3.

[36]

A. Jiménez-Casas and A. Rodríguez-Bernal, Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary, Discrete and Continuous Dynamical Systems, 39 (2019). doi: 10.3934/dcds.2019208.

[37]

T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-New York, 1976.

[38]

B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Discr. Cont. Dyn. Systems, 6 (2000), 683-690.  doi: 10.3934/dcds.2000.6.683.

[39]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33–55.

[40]

J. Lagnese, Note on the boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.  doi: 10.1137/0326068.

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J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Appl. Math., vol. 10, 1989. doi: 10.1137/1.9781611970821.

[42]

P. D. Lamberti, Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems, Complex Var. Elliptic Equ., 59 (2014), 309-323.  doi: 10.1080/17476933.2011.557155.

[43]

E. Lami Dozo and O. Torne, Symmetry and symmetry breaking for minimizers in the trace inequality, Comm. Contemp. Math., 7 (2005), 727-746.  doi: 10.1142/S0219199705001921.

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I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. 1 and 2, Cambridge University Press, 2000.

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Y. Li and M. Zhu, Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries, Comm. Pure Appl. Math., 50 (1997), 449-487.  doi: 10.1002/(SICI)1097-0312(199705)50:5<449::AID-CPA2>3.3.CO;2-5.

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J. L. Lions, Quelques Méthodes de Rèsolution des Problèmes aux Limites non Lineaires, Dunod, 1969.

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J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, RMA 8, 1988.

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J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

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A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.

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show all references

References:
[1]

R. Adams, Sobolev Spaces, Academic Press, Boston, 1978.

[2]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in, Schmeisser/Triebel: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.

[3]

G. S. Aragão and F. D. M. Bezerra, Upper semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary, J. Math. Anal. Appl., 462 (2018), 871-899.  doi: 10.1016/j.jmaa.2017.12.047.

[4]

G. S. Aragão and S. M. Bruschi, Concentrated terms and varying domains in elliptic equations: Lipschitz case, Math. Methods Appl. Sci., 39 (2016), 3450-3460.  doi: 10.1002/mma.3791.

[5]

G. S. AragãoA. L. Pereira and M. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differential Equations, 26 (2014), 871-888.  doi: 10.1007/s10884-014-9412-z.

[6]

G. S. Aragão and S. M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. Differential Equations, 253 (2012), 2573-2592.  doi: 10.1016/j.jde.2012.07.008.

[7]

G. S. AragãoA. L. Pereira and M. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Methods Appl. Sci., 35 (2012), 1110-1116.  doi: 10.1002/mma.2525.

[8]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations, 156 (1999), 376-406.  doi: 10.1006/jdeq.1998.3612.

[9]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Attractors of parabolic problems with critical nonlinearities, uniform bounds, Comm. P. D. E.'s, 25 (2000), 1-37.  doi: 10.1080/03605300008821506.

[10]

J. M. Arrieta and A. Jiménez-Casas, A. Rodríguez-Bernal, Nonhomogeneous flux condition as limit of concentrated reactions, Revista Iberoamericana de Matematicas, 24 (2008), 183–211.

[11]

J. M. Arrieta, A. Nogueira and M. C. Pereira, Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary, Discrete and Continuous Dynamical Systems, to appear.

[12]

J. M. ArrietaA. Rodríguez–Bernal and J. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary, Proceedings of The Royal Society of Edinburgh, 138A (2008), 223-237.  doi: 10.1017/S0308210506000813.

[13]

J. M. Arrieta and E. Santamaría, Distance of attractors of reaction-diffusion equations in thin domains, Journal of Differential Equations, 263 (2017), 5459-5506.  doi: 10.1016/j.jde.2017.06.023.

[14]

J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. American Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.

[15]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[16]

C. CavaterraC. GalM. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399.  doi: 10.1016/j.na.2009.11.002.

[17]

G. A. ChechkinD. CioranescuA. Damlamian and A. L. Piatnitski, On boundary value problem with singular inhomogeneity concentrated on the boundary, J. Math. Pures Appl., 98 (2012), 115-138.  doi: 10.1016/j.matpur.2011.11.002.

[18]

J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for monotone semigroups with applications to evolutionary equations, Journal of Differential Equations, 249 (2010), 485-525.  doi: 10.1016/j.jde.2010.04.006.

[19]

P. CornilleauJ. P. Loheác and A. Osses, Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers, J. of Dynamical and Control Systems, 16 (2010), 163-188.  doi: 10.1007/s10883-010-9088-6.

[20]

J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math. Z., 210 (1992), 413-439.  doi: 10.1007/BF02571805.

[21]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[22]

J. Fernández Bonder, E. Lami Dozo and J. D. Rossi, Symmetry properties for the extremals of the Sobolev trace embedding, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 21 (2004), 795–805. doi: 10.1016/j.anihpc.2003.09.005.

[23]

J. Fernández Bonder and J. D. Rossi, On the existence of extremals for the Sobolev trace embedding theorem with critical exponent, Bull. London Math. Soc., 37 (2005), 119-125.  doi: 10.1112/S0024609304003819.

[24]

C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[25]

G. GilardiA. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.  doi: 10.3934/cpaa.2009.8.881.

[26]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[27]

M. Grobbelaar-van Dalsen and N. Sauer, Solutions in Lebesgue spaces of the Navier-Stokes equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A., 123 (1993), 745-761.  doi: 10.1017/S0308210500030948.

[28]

Y. D. GolovatyD. GómezM. Lobo and E. Pérez, On vibrating membranes with very heavy thin inclusions, Math. Models Methods Appl. Sci., 14 (2004), 987-1034.  doi: 10.1142/S0218202504003520.

[29]

D. GómezM. LoboS. A. Nazarov and E. Pérez, Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues, J. Math. Pures Appl., 85 (2006), 598-632.  doi: 10.1016/j.matpur.2005.10.013.

[30]

J. K. Hale, Asymptotic Behavior of Dissipative System, 1988.

[31]

J. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326.  doi: 10.1007/BF01790353.

[32]

S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, J. of Differential Equations, 145 (1998), 184-215.  doi: 10.1006/jdeq.1997.3385.

[33]

A. Jiménez–Casas and A. Rodríguez-Bernal, Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary, Nonlinear Analysis T. M. A., 71 (2009), 2377-2383.  doi: 10.1016/j.na.2009.05.036.

[34]

A. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. and Appl., 379 (2011), 567-588.  doi: 10.1016/j.jmaa.2011.01.051.

[35]

A. Jiménez-Casas and A. Rodríguez-Bernal, Dynamic boundary conditions as a singular limit of parabolic problems with terms concentrating at the boundary, Dynamics of Partial Differential Equations, 9 (2012), 341-368.  doi: 10.4310/DPDE.2012.v9.n4.a3.

[36]

A. Jiménez-Casas and A. Rodríguez-Bernal, Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary, Discrete and Continuous Dynamical Systems, 39 (2019). doi: 10.3934/dcds.2019208.

[37]

T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-New York, 1976.

[38]

B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Discr. Cont. Dyn. Systems, 6 (2000), 683-690.  doi: 10.3934/dcds.2000.6.683.

[39]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33–55.

[40]

J. Lagnese, Note on the boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.  doi: 10.1137/0326068.

[41]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Appl. Math., vol. 10, 1989. doi: 10.1137/1.9781611970821.

[42]

P. D. Lamberti, Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems, Complex Var. Elliptic Equ., 59 (2014), 309-323.  doi: 10.1080/17476933.2011.557155.

[43]

E. Lami Dozo and O. Torne, Symmetry and symmetry breaking for minimizers in the trace inequality, Comm. Contemp. Math., 7 (2005), 727-746.  doi: 10.1142/S0219199705001921.

[44]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. 1 and 2, Cambridge University Press, 2000.

[45]

Y. Li and M. Zhu, Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries, Comm. Pure Appl. Math., 50 (1997), 449-487.  doi: 10.1002/(SICI)1097-0312(199705)50:5<449::AID-CPA2>3.3.CO;2-5.

[46]

J. L. Lions, Quelques Méthodes de Rèsolution des Problèmes aux Limites non Lineaires, Dunod, 1969.

[47]

J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, RMA 8, 1988.

[48]

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