We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments.
@article{ASNSP_2010_5_9_1_111_0, author = {De Silva, Daniela and Valdinoci, Enrico}, title = {A fully nonlinear problem with free boundary in the plane}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {111--132}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {1}, year = {2010}, mrnumber = {2668875}, zbl = {1196.35232}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2010_5_9_1_111_0/} }
TY - JOUR AU - De Silva, Daniela AU - Valdinoci, Enrico TI - A fully nonlinear problem with free boundary in the plane JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 111 EP - 132 VL - 9 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2010_5_9_1_111_0/ LA - en ID - ASNSP_2010_5_9_1_111_0 ER -
%0 Journal Article %A De Silva, Daniela %A Valdinoci, Enrico %T A fully nonlinear problem with free boundary in the plane %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 111-132 %V 9 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2010_5_9_1_111_0/ %G en %F ASNSP_2010_5_9_1_111_0
De Silva, Daniela; Valdinoci, Enrico. A fully nonlinear problem with free boundary in the plane. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 1, pp. 111-132. http://archive.numdam.org/item/ASNSP_2010_5_9_1_111_0/
[1] Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. | EuDML | MR | Zbl
and ,[2] “Fully Nonlinear Elliptic Equations”, Vol. 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995. | MR
and ,[3] “A Geometric Approach to Free Boundary Problems”, GSM 68, American Mathematical Society, Providence, Rhode Island, 2005. | MR | Zbl
and ,[4] Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D, Indiana Univ. Math. J. 58 (2009), 301–315. | MR | Zbl
and ,[5] On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003), 181–197. | EuDML | Numdam | MR | Zbl
and ,[6] Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal. 195 (2010), 1025–1058. | MR | Zbl
and ,[7] “Phase Transitions: Regularity of Flat Level Sets”, PhD thesis, University of Texas at Austin, 2003. | MR
,[8] Entire solutions to a class of fully nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2008), 369–405. | EuDML | Numdam | MR | Zbl
,[9] Regularity of flat sets in phase transitions, Ann. of Math. 169 (2009), 41–78. | MR | Zbl
,[10] Bernoulli jets and the zero mean curvature equation, J. Differential Equations 225 (2006), 710–736. | MR | Zbl
,[11] Flatness of Bernoulli jets, Math. Z. 254 (2006), 257–298. | MR | Zbl
,