Critical points of solutions to elliptic problems in planar domains

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January 2011, 10(1): 327-338. doi: 10.3934/cpaa.2011.10.327

Critical points of solutions to elliptic problems in planar domains

Jaime Arango ^1, and Adriana Gómez ^1,

1.	Departamento de Matemáticas, Universidad del Valle, Cali, Colombia, Colombia

Received April 2009 Revised October 2009 Published November 2010

Abstract
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Given a planar domain $\Omega$, and an analytic function $f$, we describe the set of critical points for the solution $u$ of the semilinear elliptic problem $\Delta u = f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. For simply connected domains we establish that the set of critical points is finite while for non--simply connected domains we show that this set is made up of finitely many isolated points and finitely many analytic Jordan curves. Further results are given in the case that $\Omega$ is an annular domain whose border has nonzero curvature.

Keywords: simply onnected domains., Morse theory, semilinear Dirichlet problems, critical points.

Mathematics Subject Classification: Primary: 74K15; Secondary: 35J05, 65M0.

Citation: Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327

References:

[1]	G. Alessandrini and R. Magnanini, The index of isolated critical points and solutions of elliptic equations in the plane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 567-589.
[2]	J. Arango, Uniqueness of critical points for semi-linear elliptic problems in convex domains, Electron. J. Differential Equations, 2005 (2005), 1-5.
[3]	J. Arango and O. Perdomo, Morse theory for analytic functions on surfaces, J. Geom., 84 (2005), 13-22. doi: doi:10.1007/s00022-005-0027-8.
[4]	V. I. Arnold, "The Theory of Singularities and its Applications,'' Cambridge University Press, Cambridge, 1991.
[5]	X. Cabré, and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.
[6]	S.Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. doi: doi:10.1007/BF02568142.
[7]	D.L. Finn, Convexity of level curves for solutions to semilinear elliptic equations, Commun. Pure Appl. Anal., 7 (2008), 1335-1343. doi: doi:10.3934/cpaa.2008.7.1335.
[8]	G. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Springer-Verlag, Berlin, 1983.
[9]	B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. Partial Differential Equations, 10 (1985), 1213-1225. doi: doi:10.1080/03605308508820404.
[10]	Xi-Nan Ma, Concavity estimates for a class of nonlinear elliptic equations in two dimensions, Math. Z., 240 (2002), 1-11. doi: doi:10.1007/s002090100341.
[11]	L.G. Makar-Limanov, Solution of Dirichlet's problem for the equation $\Delta u=-1$ in a convex region, Math. Notes Acad. Sci. U.S.S.R., 9 (1971), 52-53. doi: doi:10.1007/BF01405053.
[12]	F. Müller, On the continuation of solutions for elliptic equations in two variables, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 745-776.

show all references

References:

[1]	G. Alessandrini and R. Magnanini, The index of isolated critical points and solutions of elliptic equations in the plane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 567-589.
[2]	J. Arango, Uniqueness of critical points for semi-linear elliptic problems in convex domains, Electron. J. Differential Equations, 2005 (2005), 1-5.
[3]	J. Arango and O. Perdomo, Morse theory for analytic functions on surfaces, J. Geom., 84 (2005), 13-22. doi: doi:10.1007/s00022-005-0027-8.
[4]	V. I. Arnold, "The Theory of Singularities and its Applications,'' Cambridge University Press, Cambridge, 1991.
[5]	X. Cabré, and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.
[6]	S.Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. doi: doi:10.1007/BF02568142.
[7]	D.L. Finn, Convexity of level curves for solutions to semilinear elliptic equations, Commun. Pure Appl. Anal., 7 (2008), 1335-1343. doi: doi:10.3934/cpaa.2008.7.1335.
[8]	G. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Springer-Verlag, Berlin, 1983.
[9]	B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. Partial Differential Equations, 10 (1985), 1213-1225. doi: doi:10.1080/03605308508820404.
[10]	Xi-Nan Ma, Concavity estimates for a class of nonlinear elliptic equations in two dimensions, Math. Z., 240 (2002), 1-11. doi: doi:10.1007/s002090100341.
[11]	L.G. Makar-Limanov, Solution of Dirichlet's problem for the equation $\Delta u=-1$ in a convex region, Math. Notes Acad. Sci. U.S.S.R., 9 (1971), 52-53. doi: doi:10.1007/BF01405053.
[12]	F. Müller, On the continuation of solutions for elliptic equations in two variables, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 745-776.

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