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

Critical points of solutions to elliptic problems in planar domains

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.
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. Google Scholar

[2]

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

[3]

J. Arango and O. Perdomo, Morse theory for analytic functions on surfaces,, J. Geom., 84 (2005), 13. doi: doi:10.1007/s00022-005-0027-8. Google Scholar

[4]

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

[6]

S.Y. Cheng, Eigenfunctions and nodal sets,, Comment. Math. Helv., 51 (1976), 43. 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. 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, (1983). Google Scholar

[9]

B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?,, Comm.\ Partial Differential Equations, 10 (1985), 1213. 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. 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. 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. 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. Google Scholar

[2]

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

[3]

J. Arango and O. Perdomo, Morse theory for analytic functions on surfaces,, J. Geom., 84 (2005), 13. doi: doi:10.1007/s00022-005-0027-8. Google Scholar

[4]

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

[6]

S.Y. Cheng, Eigenfunctions and nodal sets,, Comment. Math. Helv., 51 (1976), 43. 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. 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, (1983). Google Scholar

[9]

B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?,, Comm.\ Partial Differential Equations, 10 (1985), 1213. 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. 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. 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. Google Scholar

[1]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006

[2]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[3]

Mohamed Badreddine, Thomas K. DeLillo, Saman Sahraei. A Comparison of some numerical conformal mapping methods for simply and multiply connected domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 55-82. doi: 10.3934/dcdsb.2018100

[4]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483

[5]

Jingbo Dou, Qianqiao Guo. Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains. Communications on Pure & Applied Analysis, 2012, 11 (2) : 453-464. doi: 10.3934/cpaa.2012.11.453

[6]

Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449

[7]

Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch. Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Mathematical Control & Related Fields, 2018, 8 (1) : 217-245. doi: 10.3934/mcrf.2018010

[8]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 999-1018. doi: 10.3934/dcdsb.2011.15.999

[9]

Daomin Cao, Norman E. Dancer, Ezzat S. Noussair, Shunsen Yan. On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 221-236. doi: 10.3934/dcds.1996.2.221

[10]

Fabio Giannoni, Paolo Piccione, Daniel V. Tausk. Morse theory for the travel time brachistochrones in stationary spacetimes. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 697-724. doi: 10.3934/dcds.2002.8.697

[11]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615

[12]

Keith Promislow, Hang Zhang. Critical points of functionalized Lagrangians. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1231-1246. doi: 10.3934/dcds.2013.33.1231

[13]

Grzegorz Graff, Jerzy Jezierski. Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds. Conference Publications, 2011, 2011 (Special) : 523-532. doi: 10.3934/proc.2011.2011.523

[14]

Asadollah Aghajani. Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3521-3530. doi: 10.3934/dcds.2017150

[15]

Vladimir P. Burskii, Alexei S. Zhedanov. On Dirichlet, Poncelet and Abel problems. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1587-1633. doi: 10.3934/cpaa.2013.12.1587

[16]

Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure & Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483

[17]

Wolfgang Arendt, Daniel Daners. Varying domains: Stability of the Dirichlet and the Poisson problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 21-39. doi: 10.3934/dcds.2008.21.21

[18]

Giuseppe Maria Coclite, Mario Michele Coclite. On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4923-4944. doi: 10.3934/dcds.2013.33.4923

[19]

Ping Lin. Feedback controllability for blowup points of semilinear heat equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1425-1434. doi: 10.3934/dcdsb.2017068

[20]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]