Abstract
We study the microscopic level-set convexity theorem for elliptic equation Lu = f(x, u, Du), which generalize Korevaars’ result in (Korevaar, Commun Part Diff Eq 15(4):541–556, 1990) by using different expression for the elementary symmetric functions of the principal curvatures of the level surface. We find out that the structure conditions on equation are as same as conditions in macroscopic level-set convexity results (see e.g. (Colesanti and Salani, Math Nachr 258:3–15, 2003; Greco, Bound Value Prob 1–15, 2006)). In a forthcoming paper, we use the same techniques to deal with Hessian type equations.
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Bianchini C., Longinetti M., Salani P.: Quasiconcave solutions to elliptic problems in convex rings. Indiana Univ. Math. J. 58(4), 1565–1590 (2009)
Brascamp H.J., Lieb E.H.: On extensions of the Bruun–Minkowski and Prekopa–Leindler theorems, including inequalities for log-concave functions, with an application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976)
Caffarelli L., Friedman A.: Convexity of solutions of some semilinear elliptic equations. Duke Math. J. 52, 431–455 (1985)
Caffarelli L., Spruck J.: Convexity properties of solutions to some classical variational problems. Commun. Part. Diff. Eq. 7(11), 1337–1379 (1982)
Caffarelli L., Guan P., Ma X.N.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60(12), 1769–1791 (2007)
Colesanti A., Salani P.: Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations. Math. Nachr. 258, 3–15 (2003)
Cuoghi, P., Salani, P.: Convexity of level sets for solutions to nonlinear elliptic problems in convex rings, Electron. J. Diff. Eq. 2006(124) (2006), 1C12. URL: http://ejde.math.txstate.edu
Gabriel R.: A result concerning convex level surfaces of 3-dimensional harmonic functions. J. Lond. Math. Soc. 32, 286–294 (1957)
Guan P., Ma X.N.: The Christoffel–Minkowski Problem I: Convexity of Solutions of a Hessian Equations. Inventiones Math. 151, 553–577 (2003)
Greco, A.: Quasi-concavity for semilinear elliptic equations with non-monotone and anisotrpic nonlinearities, Bound Value Prob (2006), article ID80347, 1–15
Kawhol, B.: Rearrangements and convexity of level sets in PDE. Springer Lecture Notes in Math. 1150 (1985)
Korevaar N.: Convexity of level sets for solutions to elliptic ring problems. Commun. Part. Diff. Eq. 15(4), 541–556 (1990)
Korevaar N.J., Lewis J.: Convex solutions of certain elliptic equations have constant rank hessians. Arch. Rational Mech. Anal. 91, 19–32 (1987)
Lewis J.: Capacitary functions in convex rings. Arch. Rational. Mech. Anal. 66, 201–224 (1977)
Trudinger N.S.: On new isoperimetric inequalitis and symmetrization. J. Reine Angew. Math. 488, 203–220 (1997)
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Communicated by N. Trudinger.
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Xu, L. A microscopic convexity theorem of level sets for solutions to elliptic equations. Calc. Var. 40, 51–63 (2011). https://doi.org/10.1007/s00526-010-0333-3
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DOI: https://doi.org/10.1007/s00526-010-0333-3