August  2016, 36(8): 4403-4450. doi: 10.3934/dcds.2016.36.4403

Hyperbolic boundary value problems with trihedral corners

1. 

Université Paris 13, CNRS, UMR 7539 LAGA, 99 av. Jean-Baptiste Clément, F-93430 Villetaneuse

2. 

Department of Mathematics, University of Michigan, Ann Arbor, Michigan

Received  May 2015 Revised  December 2015 Published  March 2016

Existence and uniqueness theorems are proved for boundary value problems with trihedral corners and distinct boundary conditions on the faces. Part I treats strictly dissipative boundary conditions for symmetric hyperbolic systems with elliptic or hidden elliptic generators. Part II treats the Bérenger split Maxwell equations in three dimensions with possibly discontinuous absorptions. The discontinuity set of the absorptions or their derivatives has trihedral corners. Surprisingly, there is almost no loss of derivatives for the Bérenger split problem. Both problems have their origins in numerical methods with artificial boundaries.
Citation: Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403
References:
[1]

S. Abarbanel and D. Gottlieb, A mathematical analysis of the PML method, J. Comput. Phys., 134 (1997), 357-363. doi: 10.1006/jcph.1997.5717.

[2]

A. Majda, Coercive inequalities for non elliptic symmetric systems, Comm. Pure Appl. Math., 28 (1975), 49-89.

[3]

C. Bacuta, A. L. Mazzucato, V. Nistor and L. Zikatanov, Interface and mixed boundary value problems on n-dimensional polyhedral domains, Doc. Math, 15 (2010), 687-745.

[4]

A. Benoit, Problèmes Aux Limites, Optique Géométrique et Singularités, PhD thesis, Université Nantes, 2006. https://tel.archives-ouvertes.fr/tel-01180449.

[5]

J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159.

[6]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, volume 32 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, Oxford, 2006. An introduction to rotating fluids and the Navier-Stokes equations.

[7]

F. Colombini, V. Petkov and J. Rauch, Spectral problems for non-elliptic symmetric systems with dissipative boundary conditions, Journal of Functional Analysis, 267 (2014), 1637-1661. doi: 10.1016/j.jfa.2014.06.018.

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[9]

L. Halpern, S. Petit-Bergez and J. Rauch, The analysis of matched layers, Confluentes Math., 3 (2011), 159-236. doi: 10.1142/S1793744211000291.

[10]

L. Halpern and J. Rauch, Bérenger/Maxwell with discontinuous absorptions: Existence, perfection, and no loss, Séminaire Laurent Schwartz. EDP et applications, 10, 2012-13.

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274. Springer Science & Business Media, 2007.

[12]

A. Huang and R. Temam, The linear hyperbolic initial and boundary value problems in a domain with corners, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 1627-1665. doi: 10.3934/dcdsb.2014.19.1627.

[13]

D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bulletin of the American Mathematical Society, 4 (1981), 203-207. doi: 10.1090/S0273-0979-1981-14884-9.

[14]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, Journal of Functional Analysis, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.

[15]

I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math., 24 (1971), 381-393. doi: 10.1002/cpa.3160240304.

[16]

P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13 (1960), 427-455. doi: 10.1002/cpa.3160130307.

[17]

G. Métivier and J. Rauch, Strictly dissipative nonuniqueness with corners, In Proceeding of Conference Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics, Rome. Springer, 2015. To appear.

[18]

J. Métral and O. Vacus, Caractère bien posé du problème de Cauchy pour le système de Bérenger, C. R. Math. Acad. Sci. Paris, 328 (1999), 847-852. doi: 10.1016/S0764-4442(99)80284-5.

[19]

S. J. Osher, An ill posed problem for a hyperbolic equation near a corner, Bulletin of the American Mathematical Society, 79 (1973), 1043-1044. doi: 10.1090/S0002-9904-1973-13324-5.

[20]

S. J. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. i, Transactions of the American Mathematical Society, 176 (1973), 141-164. doi: 10.1090/S0002-9947-1973-0320539-5.

[21]

S. Petit-Bergez, Problèmes Faiblement Bien Posés: Discrétisation et Applications, PhD thesis, Université Paris 13, 2006. http://tel.archives-ouvertes.fr/tel-00545794/fr/.

[22]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Transactions of the American Mathematical Society, 291 (1985), 167-187. doi: 10.1090/S0002-9947-1985-0797053-4.

[23]

J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, volume 133. American Mathematical Soc., 2012.

[24]

L. Sarason, On weak and strong solutions of boundary value problems, Com. Pure and Appl. Math, 15 (1962), 237-288. doi: 10.1002/cpa.3160150301.

[25]

L. Sarason and J. A. Smoller, Geometrical optics and the corner problem, Archive for Rational Mechanics and Analysis, 56 (1974), 34-69. doi: 10.1007/BF00279820.

[26]

M. Taniguchi, Mixed problem for wave equation in the domain with a corner, Funkcialaj Ekvacioj, 21 (1978), 249-259.

[27]

M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, volume 116 of Applied Mathematical Sciences, Springer, New York, second edition, 2011. doi: 10.1007/978-1-4419-7052-7.

show all references

References:
[1]

S. Abarbanel and D. Gottlieb, A mathematical analysis of the PML method, J. Comput. Phys., 134 (1997), 357-363. doi: 10.1006/jcph.1997.5717.

[2]

A. Majda, Coercive inequalities for non elliptic symmetric systems, Comm. Pure Appl. Math., 28 (1975), 49-89.

[3]

C. Bacuta, A. L. Mazzucato, V. Nistor and L. Zikatanov, Interface and mixed boundary value problems on n-dimensional polyhedral domains, Doc. Math, 15 (2010), 687-745.

[4]

A. Benoit, Problèmes Aux Limites, Optique Géométrique et Singularités, PhD thesis, Université Nantes, 2006. https://tel.archives-ouvertes.fr/tel-01180449.

[5]

J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159.

[6]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, volume 32 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, Oxford, 2006. An introduction to rotating fluids and the Navier-Stokes equations.

[7]

F. Colombini, V. Petkov and J. Rauch, Spectral problems for non-elliptic symmetric systems with dissipative boundary conditions, Journal of Functional Analysis, 267 (2014), 1637-1661. doi: 10.1016/j.jfa.2014.06.018.

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[9]

L. Halpern, S. Petit-Bergez and J. Rauch, The analysis of matched layers, Confluentes Math., 3 (2011), 159-236. doi: 10.1142/S1793744211000291.

[10]

L. Halpern and J. Rauch, Bérenger/Maxwell with discontinuous absorptions: Existence, perfection, and no loss, Séminaire Laurent Schwartz. EDP et applications, 10, 2012-13.

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274. Springer Science & Business Media, 2007.

[12]

A. Huang and R. Temam, The linear hyperbolic initial and boundary value problems in a domain with corners, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 1627-1665. doi: 10.3934/dcdsb.2014.19.1627.

[13]

D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bulletin of the American Mathematical Society, 4 (1981), 203-207. doi: 10.1090/S0273-0979-1981-14884-9.

[14]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, Journal of Functional Analysis, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.

[15]

I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math., 24 (1971), 381-393. doi: 10.1002/cpa.3160240304.

[16]

P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13 (1960), 427-455. doi: 10.1002/cpa.3160130307.

[17]

G. Métivier and J. Rauch, Strictly dissipative nonuniqueness with corners, In Proceeding of Conference Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics, Rome. Springer, 2015. To appear.

[18]

J. Métral and O. Vacus, Caractère bien posé du problème de Cauchy pour le système de Bérenger, C. R. Math. Acad. Sci. Paris, 328 (1999), 847-852. doi: 10.1016/S0764-4442(99)80284-5.

[19]

S. J. Osher, An ill posed problem for a hyperbolic equation near a corner, Bulletin of the American Mathematical Society, 79 (1973), 1043-1044. doi: 10.1090/S0002-9904-1973-13324-5.

[20]

S. J. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. i, Transactions of the American Mathematical Society, 176 (1973), 141-164. doi: 10.1090/S0002-9947-1973-0320539-5.

[21]

S. Petit-Bergez, Problèmes Faiblement Bien Posés: Discrétisation et Applications, PhD thesis, Université Paris 13, 2006. http://tel.archives-ouvertes.fr/tel-00545794/fr/.

[22]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Transactions of the American Mathematical Society, 291 (1985), 167-187. doi: 10.1090/S0002-9947-1985-0797053-4.

[23]

J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, volume 133. American Mathematical Soc., 2012.

[24]

L. Sarason, On weak and strong solutions of boundary value problems, Com. Pure and Appl. Math, 15 (1962), 237-288. doi: 10.1002/cpa.3160150301.

[25]

L. Sarason and J. A. Smoller, Geometrical optics and the corner problem, Archive for Rational Mechanics and Analysis, 56 (1974), 34-69. doi: 10.1007/BF00279820.

[26]

M. Taniguchi, Mixed problem for wave equation in the domain with a corner, Funkcialaj Ekvacioj, 21 (1978), 249-259.

[27]

M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, volume 116 of Applied Mathematical Sciences, Springer, New York, second edition, 2011. doi: 10.1007/978-1-4419-7052-7.

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