Convexity estimates for nonlinear elliptic equations and application to free boundary problems

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1. [1] Alt H.W., Phillips D., A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math.368 (1986) 63-107. Zbl0598.35132MR850615
2. [2] Bonnet A., Monneau R., Distribution of vortices in a type II superconductor as a free boundary problem: Existence and regularity via Nash–Moser theory, Interfaces and Free Boundaries2 (2000) 181-200. Zbl0989.35146MR1760411
3. [3] Brézis H., Kinderlehrer D., The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J.23 (9) (1974) 831-844. Zbl0278.49011MR361436
4. [4] Caffarelli L.A., Compactness method in free boundary problems, Comm. P.D.E.5 (4) (1980) 427-448. Zbl0437.35070MR567780
5. [5] Caffarelli L.A., Rivière N.M., Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa, serie IV3 (1975) 289-310. Zbl0363.35009MR412940
6. [6] Caffarelli L.A., Salazar J., Solutions of fully nonlinear elliptic equations with patches of zero gradient: existence, regularity and convexity of level curves, Trans. Amer. Math. Soc.354 (8) (2002) 3095-3115. Zbl0992.35101MR1897393
7. [7] Caffarelli L.A., Spruck J., Convexity properties of solutions to some classical variational problems, Comm. P.D.E.7 (11) (1982) 1337-1379. Zbl0508.49013MR678504
8. [8] Diaz J.I., Kawohl B., On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl.177 (1993) 263-286. Zbl0802.35087MR1224819
9. [9] Dolbeault P., Analyse Complexe, Collection Maîtrise de Mathematiques Pures, Masson, 1990. Zbl0694.30001MR1059456
10. [10] Dolbeault J., Monneau R., Estimations de convexité pour des équations elliptiques non-linéaires et application à des problèmes de frontière libre [Convexity estimates for nonlinear elliptic equations and application to free boundary problems], C. R. Acad. Sci. Paris Sér. I331 (2000) 771-776. Zbl0965.35066MR1807187
11. [11] Frehse J., On the regularity of the solution of a second order variational inequality, Boll. U.M.I.6 (4) (1972) 312-315. Zbl0261.49021MR318650
12. [12] Friedman A., Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, Wiley-Interscience, 1982. Zbl0564.49002MR679313
13. [13] Friedman A., Phillips D., The free boundary of a semilinear elliptic equation, Trans. Amer. Math. Soc.282 (1984) 153-182. Zbl0552.35079MR728708
14. [14] Hamilton R.S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc.7 (1982) 65-222. Zbl0499.58003MR656198
15. [15] Henrot A., Shahgholian H., Convexity of free boundaries with Bernoulli type boundary conditions, Nonlinear Analysis T.M.A.28 (5) (1997). Zbl0863.35117MR1422187
16. [16] Henrot A., Shahgholian H., Existence of classical solutions to a free boundary problem for the p-Laplace operator: (I) the exterior convex case, J. Reine Angew. Math.521 (2000) 85-97. Zbl0955.35078MR1752296
17. [17] Henrot A., Shahgholian H., Existence of classical solutions to a free boundary problem for the p-Laplace operator: (II) the interior convex case, Indiana Univ. Math. J.49 (1) (2000) 311-323. Zbl0977.35148MR1777029
18. [18] Kaup B., Kaup L., Holomorphic Functions of Several Variables, Walter de Gruyter, Berlin, 1983. Zbl0528.32001MR716497
19. [19] Kawohl B., When are solutions to nonlinear elliptic boundary value problems convex?, Comm. P.D.E.10 (1985) 1213-1225. Zbl0587.35026MR806439
20. [20] Kawohl B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer, 1985. Zbl0593.35002MR810619
21. [21] Kawohl B., On the convexity and symmetry of solutions to an elliptic free boundary problem, J. Austral. Math. Soc. (Series A)42 (1987) 57-68. Zbl0619.35051MR862721
22. [22] Kawohl B., On the convexity of level sets for elliptic and parabolic exterior boundary value problems, in: Potential Theory, Prague, 1987, Plenum, New York, 1988, pp. 153-159. Zbl0685.35023MR986290
23. [23] Kinderlehrer D., Nirenberg L., Regularity in free boundary problems, Bull. Amer. Math. Soc.7 (1982) 65-222. Zbl0352.35023
24. [24] Kinderlehrer D., Stampacchia G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Zbl0457.35001MR567696
25. [25] Laurence P., Stredulinsky E., Existence of regular solutions with levels for semilinear elliptic equations with nonmonotone L1 nonlinearities, Indiana Univ. Math. J.39 (4) (1990) 1081-1114. Zbl0798.35167MR1087186
26. [26] R. Monneau, Problèmes de frontières libres, EDP elliptiques non linéaires et applications en combustion, supraconductivité et élasticité, Thèse de doctorat de l'Université de Paris VI, 1999. 
27. [27] Morrey C.B., Multiple Integrals in the Calculus of Variations, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, 130, Springer-Verlag, New York, 1966. Zbl0142.38701MR202511
28. [28] Rodrigues J.F., Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987. Zbl0606.73017MR880369
29. [29] Schaeffer D.G., One-sided estimates for the curvature of the free boundary in the obstacle problem, Adv. in Math.24 (1977) 78-98. Zbl0354.35075MR448197
30. [30] Talenti G., Some estimates of solutions to Monge–Ampère type equations in dimension two, Ann. Scuola Norm. Sup. Pisa Cl. Sci., IV. Ser.8 (1981) 183-230. Zbl0467.35044MR623935

1. Jean Dolbeault, Régis Monneau, On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two