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Critical points of solutions to elliptic problems in planar domains
1. | Departamento de Matemáticas, Universidad del Valle, Cali, Colombia, Colombia |
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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