Convexity estimates for nonlinear elliptic equations and application to free boundary problems
Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 6, pp. 903-926. 
@article{AIHPC_2002__19_6_903_0,
     author = {Dolbeault, Jean and Monneau, R\'egis},
     title = {Convexity estimates for nonlinear elliptic equations and application to free boundary problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {903--926},
     publisher = {Elsevier},
     volume = {19},
     number = {6},
     year = {2002},
     mrnumber = {1939090},
     zbl = {1034.35047},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPC_2002__19_6_903_0/}
}
TY  - JOUR
AU  - Dolbeault, Jean
AU  - Monneau, Régis
TI  - Convexity estimates for nonlinear elliptic equations and application to free boundary problems
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2002
SP  - 903
EP  - 926
VL  - 19
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/item/AIHPC_2002__19_6_903_0/
LA  - en
ID  - AIHPC_2002__19_6_903_0
ER  - 
%0 Journal Article
%A Dolbeault, Jean
%A Monneau, Régis
%T Convexity estimates for nonlinear elliptic equations and application to free boundary problems
%J Annales de l'I.H.P. Analyse non linéaire
%D 2002
%P 903-926
%V 19
%N 6
%I Elsevier
%U http://archive.numdam.org/item/AIHPC_2002__19_6_903_0/
%G en
%F AIHPC_2002__19_6_903_0
Dolbeault, Jean; Monneau, Régis. Convexity estimates for nonlinear elliptic equations and application to free boundary problems. Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 6, pp. 903-926. http://archive.numdam.org/item/AIHPC_2002__19_6_903_0/

[1] Alt H.W., Phillips D., A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986) 63-107. | MR | Zbl

[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 Boundaries 2 (2000) 181-200. | MR | Zbl

[3] Brézis H., Kinderlehrer D., The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (9) (1974) 831-844. | MR | Zbl

[4] Caffarelli L.A., Compactness method in free boundary problems, Comm. P.D.E. 5 (4) (1980) 427-448. | MR | Zbl

[5] Caffarelli L.A., Rivière N.M., Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa, serie IV 3 (1975) 289-310. | Numdam | MR | Zbl

[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. | MR | Zbl

[7] Caffarelli L.A., Spruck J., Convexity properties of solutions to some classical variational problems, Comm. P.D.E. 7 (11) (1982) 1337-1379. | MR | Zbl

[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. | MR | Zbl

[9] Dolbeault P., Analyse Complexe, Collection Maîtrise de Mathematiques Pures, Masson, 1990. | MR | Zbl

[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. I 331 (2000) 771-776. | MR | Zbl

[11] Frehse J., On the regularity of the solution of a second order variational inequality, Boll. U.M.I. 6 (4) (1972) 312-315. | MR | Zbl

[12] Friedman A., Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, Wiley-Interscience, 1982. | MR | Zbl

[13] Friedman A., Phillips D., The free boundary of a semilinear elliptic equation, Trans. Amer. Math. Soc. 282 (1984) 153-182. | MR | Zbl

[14] Hamilton R.S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982) 65-222. | MR | Zbl

[15] Henrot A., Shahgholian H., Convexity of free boundaries with Bernoulli type boundary conditions, Nonlinear Analysis T.M.A. 28 (5) (1997). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[18] Kaup B., Kaup L., Holomorphic Functions of Several Variables, Walter de Gruyter, Berlin, 1983. | MR | Zbl

[19] Kawohl B., When are solutions to nonlinear elliptic boundary value problems convex?, Comm. P.D.E. 10 (1985) 1213-1225. | MR | Zbl

[20] Kawohl B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer, 1985. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[23] Kinderlehrer D., Nirenberg L., Regularity in free boundary problems, Bull. Amer. Math. Soc. 7 (1982) 65-222.

[24] Kinderlehrer D., Stampacchia G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. | MR | Zbl

[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. | MR | Zbl

[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] Morrey C.B., Multiple Integrals in the Calculus of Variations, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, 130, Springer-Verlag, New York, 1966. | MR | Zbl

[28] Rodrigues J.F., Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl