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 & Continuous Dynamical Systems - A, 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. doi: 10.1006/jcph.1997.5717. Google Scholar

[2]

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

[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. Google Scholar

[4]

A. Benoit, Problèmes Aux Limites, Optique Géométrique et Singularités,, PhD thesis, (2006). Google Scholar

[5]

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

[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, (2006). Google Scholar

[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. doi: 10.1016/j.jfa.2014.06.018. Google Scholar

[8]

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

[9]

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

[10]

L. Halpern and J. Rauch, Bérenger/Maxwell with discontinuous absorptions: Existence, perfection, and no loss,, Séminaire Laurent Schwartz. EDP et applications, (): 2012. Google Scholar

[11]

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

[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. doi: 10.3934/dcdsb.2014.19.1627. Google Scholar

[13]

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

[14]

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

[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. doi: 10.1002/cpa.3160240304. Google Scholar

[16]

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

[17]

G. Métivier and J. Rauch, Strictly dissipative nonuniqueness with corners,, In Proceeding of Conference Shocks, (2015). Google Scholar

[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. doi: 10.1016/S0764-4442(99)80284-5. Google Scholar

[19]

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

[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. doi: 10.1090/S0002-9947-1973-0320539-5. Google Scholar

[21]

S. Petit-Bergez, Problèmes Faiblement Bien Posés: Discrétisation et Applications,, PhD thesis, (2006). Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, volume 116 of Applied Mathematical Sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7052-7. Google Scholar

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[4]

A. Benoit, Problèmes Aux Limites, Optique Géométrique et Singularités,, PhD thesis, (2006). Google Scholar

[5]

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

[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, (2006). Google Scholar

[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. doi: 10.1016/j.jfa.2014.06.018. Google Scholar

[8]

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

[9]

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

[10]

L. Halpern and J. Rauch, Bérenger/Maxwell with discontinuous absorptions: Existence, perfection, and no loss,, Séminaire Laurent Schwartz. EDP et applications, (): 2012. Google Scholar

[11]

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

[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. doi: 10.3934/dcdsb.2014.19.1627. Google Scholar

[13]

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

[14]

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

[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. doi: 10.1002/cpa.3160240304. Google Scholar

[16]

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

[17]

G. Métivier and J. Rauch, Strictly dissipative nonuniqueness with corners,, In Proceeding of Conference Shocks, (2015). Google Scholar

[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. doi: 10.1016/S0764-4442(99)80284-5. Google Scholar

[19]

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

[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. doi: 10.1090/S0002-9947-1973-0320539-5. Google Scholar

[21]

S. Petit-Bergez, Problèmes Faiblement Bien Posés: Discrétisation et Applications,, PhD thesis, (2006). Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, volume 116 of Applied Mathematical Sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7052-7. Google Scholar

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