# American Institute of Mathematical Sciences

January  2011, 10(1): 225-243. doi: 10.3934/cpaa.2011.10.225

## Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$

 1 Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui Province, China, China

Received  December 2009 Revised  April 2010 Published  November 2010

We give a sharp lower bound for the principal curvature of the level sets of harmonic functions and minimal graphs defined on convex rings in $R^3$ with homogeneous Dirichlet boundary conditions.
Citation: Xi-Nan Ma, Jiang Ye, Yun-Hua Ye. Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$. Communications on Pure and Applied Analysis, 2011, 10 (1) : 225-243. doi: 10.3934/cpaa.2011.10.225
##### References:
 [1] L. V. Ahlfors, "Conformal Invariants: Topics in Geometric Function Theory," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973 (pp 5-6). [2] B. J. Bian, P. Guan, X. N. Ma and L. Xu, A microscopic convexity principle for the level sets of solution for nonlinear elliptic partial differential equations,, to appear in Indiana Univ. Math. J.., (). [3] C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math. J., 58 (2009), 1565-1589. doi: doi:10.1512/iumj.2009.58.3539. [4] L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52 (1985), 431-456. doi: doi:10.1215/S0012-7094-85-05221-4. [5] L. A. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations IV: Starshaped compact Weigarten hypersurfaces, in "Current Topics in Partial Differential Equations," Y. Ohya, K. Kasahara and N. Shimakura (eds), Kinokuniya, Tokyo, 1985, 1-26. [6] L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differ. Equations, 7 (1982), 1337-1379. doi: doi:10.1080/03605308208820254. [7] A. Chang, X. N. Ma and P. Yang, Principal curvature estimates for the convex level sets of semilinear elliptic equations, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1151-1164. [8] J. Dolbeault and R. Monneau, Convexity estimates for nonlinear elliptic equations and application to free boundary problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 903-926. doi: doi:10.1016/S0294-1449(02)00106-3. [9] R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc., 32 (1957), 286-294. doi: doi:10.1112/jlms/s1-32.3.286. [10] J. J. Gergen, Note on the Green function of a star-shaped three dimensional region, Amer. J. Math., 53 (1931), 746-752. doi: doi:10.2307/2371223. [11] S. Gleason and T. Wolff, Lewy's harmonic gradient maps in higher dimensions, Comm. Partial Diff. Equations, 16 (1991), 1925-1968. doi: doi:10.1080/03605309108820828. [12] J. Jost, X. N. Ma and Q. Z. Ou, Curvature estimates in dimensions 2 and 3 for the level sets of p-harmonic functions in convex rings,, to appear in Trans. Amer. Math. Soc.., (). [13] B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lectures Notes in Math., 1150, Springer-Verlag, Berlin, 1985. [14] N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differ. Equations, 15 (1990), 541-556. doi: doi:10.1080/03605309908820698. [15] J. L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224. doi: doi:10.1007/BF00250671. [16] H. Lewy, On the non-vanishing of the Jacobian of a homeomorphism by harmonic Gradients, Annals of Math., 88 (1968), 518-529. doi: doi:10.2307/1970723. [17] M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A, 6 (1983), 71-75. [18] M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes, J. Diff. Equations, 67 (1987), 344-358. doi: doi:10.1016/0022-0396(87)90131-8. [19] X. N. Ma, Q. Z. Ou and W. Zhang, Gaussian Curvature estimates for the convex level sets of $p$-harmonic functions, Comm. Pure Appl. Math., 63 (2010), no. 7, 935-971. [20] M. Ortel and W. Schneider, Curvature of level curves of harmonic functions, Canad. Math. Bull., 26 (1983), 399-405. [21] J. P. Rosay and W. Rudin, A maximum principle for sums of subharmonic functions,and the convexity of level sets, Michigan Math. J., 36 (1989), 95-111. doi: doi:10.1307/mmj/1029003884. [22] M. Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Annals of Math., 63 (1956), 77-90. doi: doi:10.2307/1969991. [23] G. Talenti, On functions, whoselines of steepest descent bend proportionally to level lines, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 587-605.

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##### References:
 [1] L. V. Ahlfors, "Conformal Invariants: Topics in Geometric Function Theory," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973 (pp 5-6). [2] B. J. Bian, P. Guan, X. N. Ma and L. Xu, A microscopic convexity principle for the level sets of solution for nonlinear elliptic partial differential equations,, to appear in Indiana Univ. Math. J.., (). [3] C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math. J., 58 (2009), 1565-1589. doi: doi:10.1512/iumj.2009.58.3539. [4] L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52 (1985), 431-456. doi: doi:10.1215/S0012-7094-85-05221-4. [5] L. A. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations IV: Starshaped compact Weigarten hypersurfaces, in "Current Topics in Partial Differential Equations," Y. Ohya, K. Kasahara and N. Shimakura (eds), Kinokuniya, Tokyo, 1985, 1-26. [6] L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differ. Equations, 7 (1982), 1337-1379. doi: doi:10.1080/03605308208820254. [7] A. Chang, X. N. Ma and P. Yang, Principal curvature estimates for the convex level sets of semilinear elliptic equations, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1151-1164. [8] J. Dolbeault and R. Monneau, Convexity estimates for nonlinear elliptic equations and application to free boundary problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 903-926. doi: doi:10.1016/S0294-1449(02)00106-3. [9] R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc., 32 (1957), 286-294. doi: doi:10.1112/jlms/s1-32.3.286. [10] J. J. Gergen, Note on the Green function of a star-shaped three dimensional region, Amer. J. Math., 53 (1931), 746-752. doi: doi:10.2307/2371223. [11] S. Gleason and T. Wolff, Lewy's harmonic gradient maps in higher dimensions, Comm. Partial Diff. Equations, 16 (1991), 1925-1968. doi: doi:10.1080/03605309108820828. [12] J. Jost, X. N. Ma and Q. Z. Ou, Curvature estimates in dimensions 2 and 3 for the level sets of p-harmonic functions in convex rings,, to appear in Trans. Amer. Math. Soc.., (). [13] B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lectures Notes in Math., 1150, Springer-Verlag, Berlin, 1985. [14] N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differ. Equations, 15 (1990), 541-556. doi: doi:10.1080/03605309908820698. [15] J. L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224. doi: doi:10.1007/BF00250671. [16] H. Lewy, On the non-vanishing of the Jacobian of a homeomorphism by harmonic Gradients, Annals of Math., 88 (1968), 518-529. doi: doi:10.2307/1970723. [17] M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A, 6 (1983), 71-75. [18] M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes, J. Diff. Equations, 67 (1987), 344-358. doi: doi:10.1016/0022-0396(87)90131-8. [19] X. N. Ma, Q. Z. Ou and W. Zhang, Gaussian Curvature estimates for the convex level sets of $p$-harmonic functions, Comm. Pure Appl. Math., 63 (2010), no. 7, 935-971. [20] M. Ortel and W. Schneider, Curvature of level curves of harmonic functions, Canad. Math. Bull., 26 (1983), 399-405. [21] J. P. Rosay and W. Rudin, A maximum principle for sums of subharmonic functions,and the convexity of level sets, Michigan Math. J., 36 (1989), 95-111. doi: doi:10.1307/mmj/1029003884. [22] M. Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Annals of Math., 63 (1956), 77-90. doi: doi:10.2307/1969991. [23] G. Talenti, On functions, whoselines of steepest descent bend proportionally to level lines, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 587-605.
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