Abstract
Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.
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Aftalion A., Busca J., Reichel W.: Approximate radial symmetry for overdetermined boundary value problems. Adv. Differ. Equ. 4(6), 907–932 (1999)
Alvino, A., Ferone, V., Nitsch, C.: A sharp isoperimetric inequality in the plane 2008, Preprint
Berestycki H., Nirenberg L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22(1), 1–37 (1991)
Bonnesen, T.: Über eine Verschärfung der isoperimetrischen Ungleichheit des Kreises in der Ebene und auf der Kugeloberfläche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körner. Math. Ann. 84(3–4), 216–227 (1921)
Brandolini, B., Nitsch, C., Salani, P., Trombetti, C.: Serrin type overdetermined problems: an alternative proof. Arch. Rat. Mech. Anal. (to appear). doi:10.1007/s00205-008-0119-3
Brandolini B., Nitsch C., Salani P., Trombetti C.: On the stability of the Serrin problem. J. Differ. Equ. 245(6), 1566–1583 (2008)
Burago Yu.D., Zalgaller V.A.: Geometric inequalities. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285. Springer Series in Soviet Mathematics. Springer, Berlin (1988)
Caffarelli L.A., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37(3), 369–402 (1984)
Caffarelli L.A., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)
Fraenkel L.E.: An introduction to maximum principles and symmetry in elliptic problems. Cambridge Tracts in Mathematics, vol. 128. Cambridge University Press, Cambridge (2000)
Fuglede B.: Stability in the isoperimetric problem for convex or nearly spherical domains in \({\mathbb {R}^n}\). Trans. Am. Math. Soc. 314(2), 619–638 (1989)
Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 1–40 (2008)
Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68(3), 209–243 (1979)
Groemer H.: Stability for geometric inequalities, Handbook of Convex Geometry, vol. A, pp. 125–150
Groemer H., Schneider R.: Stability estimates for some geometric inequalities. Bull London Math. Soc. 23(1), 67–74 (1991)
Henrot A., Philippin G.A.: Approximate of radial symmetry for solutions of a class of boundary value problems in ring-shaped domains. Z. Angew. Math. Phys. 54(5), 784–796 (2003)
Rockafellar R.T.: Convex analysis. Princeton Mathematical Series, 28. Princeton University Press, Princeton, NJ (1970)
Rosset E.: An approximate Gidas-Ni-Nirenberg theorem. Math. Methods Appl. Sci. 17(13), 1045–1052 (1994)
Serrin J.: A symmetry problem in potential theory. Arch. Rat. Mech. Anal. 43, 304–318 (1971)
Schneider R.: Convex bodies: the Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge (1993)
Villani C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)
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Brandolini, B., Nitsch, C., Salani, P. et al. Stability of radial symmetry for a Monge-Ampère overdetermined problem. Annali di Matematica 188, 445–453 (2009). https://doi.org/10.1007/s10231-008-0083-4
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DOI: https://doi.org/10.1007/s10231-008-0083-4