A maximum principle for the system Δu − ∇W(u)=0

Panagiotis Antonopoulos; Panayotis Smyrnelis

doi:10.1016/j.crma.2016.03.015

Partial differential equations/Calculus of variations

A maximum principle for the system Δu − ∇W(u)=0
[Un principe du maximum pour le système Δu − ∇W(u)=0]

Présenté par : Haïm Brézis

Panagiotis Antonopoulos ¹ ; Panayotis Smyrnelis ²

¹ Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece
² Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 595-600.

Résumé
Abstract

Nous établissons un principe du maximum pour les solutions minimales du système $Δ u - \nabla W (u) = 0$ , dont le potentiel W s'annule à la frontière d'un ensemble fermé convexe $C_{0} \subset R^{m}$ , de classe $C^{2}$ ou réduit à un point ${a}$ .

A maximum principle is established for minimal solutions to the system $Δ u - \nabla W (u) = 0$ , with a potential W vanishing at the boundary of a closed convex set $C_{0} \subset R^{m}$ , which is either $C^{2}$ smooth or coincides with a point ${a}$ .

Reçu le : 2015-08-05
Accepté le : 2016-04-01
Publié le : 2016-04-14

DOI : 10.1016/j.crma.2016.03.015

Affiliations des auteurs :

Panagiotis Antonopoulos ¹ ; Panayotis Smyrnelis ²

¹ Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece
² Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile

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     author = {Panagiotis Antonopoulos and Panayotis Smyrnelis},
     title = {A maximum principle for the system {\ensuremath{\Delta}\protect\emph{u}\,\ensuremath{-}\,\ensuremath{\nabla}\protect\emph{W}(\protect\emph{u})=0}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {595--600},
     publisher = {Elsevier},
     volume = {354},
     number = {6},
     year = {2016},
     doi = {10.1016/j.crma.2016.03.015},
     language = {en},
}

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Panagiotis Antonopoulos; Panayotis Smyrnelis. A maximum principle for the system Δu − ∇W(u)=0. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 595-600. doi : 10.1016/j.crma.2016.03.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.015/

Bibliographie
Cité par

[1] N.D. Alikakos; G. Fusco A maximum principle for systems with variational structure and an application to standing waves, J. Eur. Math. Soc., Volume 17 (2015) no. 7, pp. 1547-1567

[2] S. Baldo Minimal interface criterion for phase transitions in mixtures of Cahn–Hilliard fluids, Ann. Inst. Henri Poincaré, Volume 7 (1990) no. 2, pp. 67-90

[3] J.M. Ball; E.C.M. Crooks Local minimizers and planar interfaces in a phase-transition model with interfacial energy, Calc. Var. Partial Differ. Equ., Volume 40 (2011) no. 3–4, pp. 501-538

[4] H. Brézis Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag, New York, 2011

[5] R.G. Casten; C.J. Holland Instability results for reaction-diffusion equations with Neumann boundary conditions, J. Differ. Equ., Volume 27 (1978), pp. 266-273

[6] L.C. Evans A strong maximum principle for parabolic systems in a convex set with arbitrary boundary, Proc. Amer. Math. Soc., Volume 138 (2010) no. 9, pp. 3179-3185

[7] L.C. Evans Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 2010

[8] L.C. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, USA, 1992

[9] R.V. Kohn; P. Sternberg Local minimisers and singular perturbations, Proc. R. Soc. Edinb., Sect. A, Volume 111 (1989) no. 1–2, pp. 69-84

[10] H. Matano Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., Volume 15 (1979) no. 2, pp. 401-454

[11] L. Modica; S. Mortola Un esempio di Γ-convergenza, Boll. Unione Mat. Ital., B, Volume 14 (1977) no. 1, pp. 285-299

[12] H. Weinberger Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. Ser. VI, Volume 8 (1975), pp. 295-310

[13] H. Weinberger Some remarks on invariant sets for systems (P.W. Schaefer, ed.), Maximum Principles and Eigenvalue Problems in Partial Differential Equations, Pitman Research Notes in Mathematics, vol. 175, Longman, 1988, pp. 189-207

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