Artículo Original
Publicado 2017-08-08
Palabras clave
- Principio del máximo,
- puntos críticos,
- componentes conexas,
- simetría
Cómo citar
Arango, J., Jiménez, J., & Salazar, A. (2017). Puntos críticos y simetrías en problemas elípticos. Revista Integración, Temas De matemáticas, 35(1), 1–9. https://doi.org/10.18273/revint.v35n1-2017001
Resumen
Se estima una cota superior para el número de puntos críticos de la solución de un problema semilineal elíptico con condición de Dirichlet nula en el borde de un dominio planar. El resultado se obtiene en dominios simétricos con respecto a una recta y convexos en la dirección ortogonal a la misma.
MSC2010: 35J25, 35J91, 74K15.
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Referencias
[1] Alessandrini G., “Critical points of solutions of elliptic equations in two variables”, Ann.Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (1987), No. 2, 229–256.
[2] Arango J. and Gómez A., “Critical points of solutions to elliptic problems in planar domains”, Commun. Pure Appl. Anal. 10 (2011), No. 1, 327–338.
[3] Arango J. and Gómez A., “Critical points of solutions to quasilinear elliptic problems”, Nonlinear Anal. 75 (2012), No. 11, 4375–4381.
[4] Cabré X. and Chanillo S., “Stable solutions of semilinear elliptic problems in convex domains”, Selecta Math. (N.S.) 4 (1998), No. 1, 1–10.
[5] Caffarelli L. and Spruck J., “Convexity properties of solutions to some classical variational problems”, Comm. Partial Differential Equations 7 (1982), No. 11, 1337–1379.
[6] Cheng S., “Eigenfunctions and nodal sets”, Comment. Math. Helv. 51 (1976), No. 1, 43–55.
[7] Chipot M., Elliptic equations: an introductory course, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 2009.
[8] Duren P., Harmonic mappings in the plane, Cambridge Univ. Press, Cambridge, 2004.
[9] Finn D., “Convexity of level curves for solutions to semilinear elliptic equations”, Commun. Pure Appl. Anal. 7 (2008), No. 6, 1335–1343.
[10] Greco A., “Extremality conditions for the quasi-concavity function and applications”, Arch. Math. (Basel) 93 (2009), No. 4, 389–398.
[11] Kawohl B., “When are solutions to nonlinear elliptic boundary value problems convex?”, Comm. Partial Differential Equations 10 (1985), No. 10, 1213–1225.
[12] Makar-Limanov L., “Solution of Dirichlet’s problem for the equation u = −1 in a convex region”, Math. Notes 9 (1971), No. 1, 52–53.
[13] Müller F., “On the continuation of solutions for elliptic equations in two variables”, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), No. 6, 745–776.
[14] Rivera A., “Puntos críticos de soluciones de problemas elípticos con condición de Dirichlet”, Tesis de maestría, Universidad del Valle, Cali, 2006, 57 p.
[15] Sakaguchi S., “Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (1987), No. 3, 403–421.
[16] Pucci P. and Serrin J., The maximum principle, Birkhäuser Verlag, Basel, 2007.
[2] Arango J. and Gómez A., “Critical points of solutions to elliptic problems in planar domains”, Commun. Pure Appl. Anal. 10 (2011), No. 1, 327–338.
[3] Arango J. and Gómez A., “Critical points of solutions to quasilinear elliptic problems”, Nonlinear Anal. 75 (2012), No. 11, 4375–4381.
[4] Cabré X. and Chanillo S., “Stable solutions of semilinear elliptic problems in convex domains”, Selecta Math. (N.S.) 4 (1998), No. 1, 1–10.
[5] Caffarelli L. and Spruck J., “Convexity properties of solutions to some classical variational problems”, Comm. Partial Differential Equations 7 (1982), No. 11, 1337–1379.
[6] Cheng S., “Eigenfunctions and nodal sets”, Comment. Math. Helv. 51 (1976), No. 1, 43–55.
[7] Chipot M., Elliptic equations: an introductory course, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 2009.
[8] Duren P., Harmonic mappings in the plane, Cambridge Univ. Press, Cambridge, 2004.
[9] Finn D., “Convexity of level curves for solutions to semilinear elliptic equations”, Commun. Pure Appl. Anal. 7 (2008), No. 6, 1335–1343.
[10] Greco A., “Extremality conditions for the quasi-concavity function and applications”, Arch. Math. (Basel) 93 (2009), No. 4, 389–398.
[11] Kawohl B., “When are solutions to nonlinear elliptic boundary value problems convex?”, Comm. Partial Differential Equations 10 (1985), No. 10, 1213–1225.
[12] Makar-Limanov L., “Solution of Dirichlet’s problem for the equation u = −1 in a convex region”, Math. Notes 9 (1971), No. 1, 52–53.
[13] Müller F., “On the continuation of solutions for elliptic equations in two variables”, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), No. 6, 745–776.
[14] Rivera A., “Puntos críticos de soluciones de problemas elípticos con condición de Dirichlet”, Tesis de maestría, Universidad del Valle, Cali, 2006, 57 p.
[15] Sakaguchi S., “Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (1987), No. 3, 403–421.
[16] Pucci P. and Serrin J., The maximum principle, Birkhäuser Verlag, Basel, 2007.