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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

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 & 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.  Google Scholar

[2]

J. Arango, Uniqueness of critical points for semi-linear elliptic problems in convex domains, Electron. J. Differential Equations, 2005 (2005), 1-5.  Google Scholar

[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.  Google Scholar

[4]

V. I. Arnold, "The Theory of Singularities and its Applications,'' Cambridge University Press, Cambridge, 1991.  Google Scholar

[5]

X. Cabré, and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.  Google Scholar

[6]

S.Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. doi: doi:10.1007/BF02568142.  Google Scholar

[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.  Google Scholar

[8]

G. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Springer-Verlag, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Arango, Uniqueness of critical points for semi-linear elliptic problems in convex domains, Electron. J. Differential Equations, 2005 (2005), 1-5.  Google Scholar

[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.  Google Scholar

[4]

V. I. Arnold, "The Theory of Singularities and its Applications,'' Cambridge University Press, Cambridge, 1991.  Google Scholar

[5]

X. Cabré, and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.  Google Scholar

[6]

S.Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. doi: doi:10.1007/BF02568142.  Google Scholar

[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.  Google Scholar

[8]

G. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Springer-Verlag, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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