调和函数和极小曲面凸水平集的一些曲率估计
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摘要
本文讨论三维调和函数、三维极小图凸水平集的最小主曲率的估计，浸入极小曲面凸水平集的高斯曲率估计以及Hessian型方程允许解的对数梯度估计
全文共分六章，第一章回顾椭圆方程解的水平集的凸性的历史；第二章引入刻画函数水平集主曲率的曲率矩阵（？）；第三、四章利用该曲率矩阵证明了如下一些定理：
定理0.1设Ω为R~n中的区域，3≤n≤5且u为Ω上的p-调和函数，即设在（？），（？）,u的水平集关于梯度方向（？）为严格凸的，则u的水平集的主曲率不能在Ω的内部达到最小值，除非u是常数
利用凸环上p-调和函数的水平集是严格凸的，可以得到下面的推论
推论0.2设u为下面边值问题的解，其中Ω_0,Ω_1为R~n中的有界凸集，满足若3 对于极小图情形有
定理0.3设为的区域，u为Ω上的极小图，即u满足极小曲面方程假设在Ω上。若u水平集关于梯度方向(?)为严格凸的，则u的水平集的主曲率不能在Ω内部达到极小，除非u是常数
利用凸环上的极小图的水平集是严格凸的，同样可以得到下面的推论推论0.4设u满足其中Ω_0和Ω_1为。中满足的有界凸区域。则的水平集的主曲率在的边界上达到极小值
第五章利用活动标架计算得到了浸入极小曲面凸水平集高斯曲率的估计
定理0.5设为浸入极小曲面，是关于方向∈的高度函数，并且没有临界点，如果u的所有水平集关于方向为严格凸的，那么对于或，函数不能在的内部取得最小值，除非它为常数
第六章考虑Hessian型方程允许光滑解的对数梯度估计
定理0.6设为下面Hessian型方程的允许光滑解
In this dissertation, we discuss the smallest principal curvature estimates of convex level sets of three-dimensional harmonic function and minimal graph, Gaussian curvature estimates of convex level sets of minimal surface. Logarithmic gradient estimates of admissible solution to Hessian type equation arising in geometric optics is also considered. The dissertation consists of six chapters.
In chapter 1, we briefly recall the history of convex level sets of solutions to elliptic equations. In chapter 2, we introduce the symmetric curvature matrix {aij} to describe the principal curvatures of level sets of a function. In chapter 3 and 4, we use this curvature matrix to prove the following results:
Theorem 1 Let be a bounded smooth domain, 3 < n < 5 and u be a p-harmonic function inΩ, i.e.Assume in < p < 3 and the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ，unless it is constant.
Using the fact that the level sets of p-harmonic function over convex ring are strictly convex, we have
Corollary 0.1 Let u be the solution of the following boundary value problem,where and are two bounded convex sets in satisfying C If 3 < n < 5 and，then the principal curvature of the level sets of u attains its minimum on .
For the case of the three-dimensional minimal graph, we have
Theorem 2 Let be a domain in and u be a minimal graph over，i.e., u satisfy the minimal surface equationAssume in . If the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ, unless it is constant.Using the fact that the level sets of the minimal graph over convex ring are strictly convex , we get the following
Corollary 0.2 Let u satisfywhere and are bounded convex domains in R3; satisfying Then the principal curvature of the level sets of u attains its minimum value on d .In chapter 5, we discuss the Gaussian curvature estimates of convex level sets of immersed minimal surface using moving frame method and obtain the following
Theorem 3 Let be an immersed minimal surface. Let u be the height function of Mn corresponding to the direction .Ifu has no critical point, and let the level sets of u are all strictly convex with respect to direction Vu. Denote the Gaussian curvature by K and let . Then the function can not attain its minimum at an interior point of Mn, unless it is constant.
In chapter 6, we give logarithmic gradient estimate of admissible solution to Hessian type equation and obtain the following
Theorem 4 be a smooth admissible solution to the following equationwhereLet f be a positive function. Then we have
全文共分六章，第一章回顾椭圆方程解的水平集的凸性的历史；第二章引入刻画函数水平集主曲率的曲率矩阵（？）；第三、四章利用该曲率矩阵证明了如下一些定理：
定理0.1设Ω为R~n中的区域，3≤n≤5且u为Ω上的p-调和函数，即设在（？），（？）,u的水平集关于梯度方向（？）为严格凸的，则u的水平集的主曲率不能在Ω的内部达到最小值，除非u是常数
利用凸环上p-调和函数的水平集是严格凸的，可以得到下面的推论
推论0.2设u为下面边值问题的解，其中Ω_0,Ω_1为R~n中的有界凸集，满足若3
定理0.3设为的区域，u为Ω上的极小图，即u满足极小曲面方程假设在Ω上。若u水平集关于梯度方向(?)为严格凸的，则u的水平集的主曲率不能在Ω内部达到极小，除非u是常数
利用凸环上的极小图的水平集是严格凸的，同样可以得到下面的推论推论0.4设u满足其中Ω_0和Ω_1为。中满足的有界凸区域。则的水平集的主曲率在的边界上达到极小值
第五章利用活动标架计算得到了浸入极小曲面凸水平集高斯曲率的估计
定理0.5设为浸入极小曲面，是关于方向∈的高度函数，并且没有临界点，如果u的所有水平集关于方向为严格凸的，那么对于或，函数不能在的内部取得最小值，除非它为常数
第六章考虑Hessian型方程允许光滑解的对数梯度估计
定理0.6设为下面Hessian型方程的允许光滑解
In this dissertation, we discuss the smallest principal curvature estimates of convex level sets of three-dimensional harmonic function and minimal graph, Gaussian curvature estimates of convex level sets of minimal surface. Logarithmic gradient estimates of admissible solution to Hessian type equation arising in geometric optics is also considered. The dissertation consists of six chapters.
In chapter 1, we briefly recall the history of convex level sets of solutions to elliptic equations. In chapter 2, we introduce the symmetric curvature matrix {aij} to describe the principal curvatures of level sets of a function. In chapter 3 and 4, we use this curvature matrix to prove the following results:
Theorem 1 Let be a bounded smooth domain, 3 < n < 5 and u be a p-harmonic function inΩ, i.e.Assume in < p < 3 and the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ，unless it is constant.
Using the fact that the level sets of p-harmonic function over convex ring are strictly convex, we have
Corollary 0.1 Let u be the solution of the following boundary value problem,where and are two bounded convex sets in satisfying C If 3 < n < 5 and，then the principal curvature of the level sets of u attains its minimum on .
For the case of the three-dimensional minimal graph, we have
Theorem 2 Let be a domain in and u be a minimal graph over，i.e., u satisfy the minimal surface equationAssume in . If the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ, unless it is constant.Using the fact that the level sets of the minimal graph over convex ring are strictly convex , we get the following
Corollary 0.2 Let u satisfywhere and are bounded convex domains in R3; satisfying Then the principal curvature of the level sets of u attains its minimum value on d .In chapter 5, we discuss the Gaussian curvature estimates of convex level sets of immersed minimal surface using moving frame method and obtain the following
Theorem 3 Let be an immersed minimal surface. Let u be the height function of Mn corresponding to the direction .Ifu has no critical point, and let the level sets of u are all strictly convex with respect to direction Vu. Denote the Gaussian curvature by K and let . Then the function can not attain its minimum at an interior point of Mn, unless it is constant.
In chapter 6, we give logarithmic gradient estimate of admissible solution to Hessian type equation and obtain the following
Theorem 4 be a smooth admissible solution to the following equationwhereLet f be a positive function. Then we have
引文
[1] Ahlfors L.V., Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Djsseldorf-Johannesburg, 1973(pp 5–6).
[2] B.J. Bian, P. Guan, X.N. Ma and L. Xu, A microscopic convexity principle for thelevel sets of solution for nonlinear elliptic partial differential equations, to appearin Indiana Univ. Math. J..
[3] B.J.Bian and P.F.Guan, A microscopic convexity principle for nonlinear differentialequations, Invent. Math. 177 (2009), no. 2, 307–335.
[4] C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic prob-lems in convex rings, Indiana Univ. Math. J. 58 (2009), no. 4, 1565–1589.
[5] L.A. Caffarelli and A. Friedman, Convexity of solutions of some semilinear ellipticequations, Duke Math. J., 52, (1985), 431–455.
[6] L.A. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equa-tions IV: Starshaped compact Weigarten hypersurfaces, Current topics in partialdifferential equations, 1–26, Kinokuniya, Tokyo, 1986.
[7] L.A. Caffarelli and J. Spruck, Convexity properties of solutions to some classicalvariational problems, Comm. Partial Differ. Equations, 7, (1982), 1337–1379.
[8] L.A.Caffarelli and V.I.Oliker, Weak solutions of one inverse problem in geoetricoptics, J. Math. Sci. (N. Y.) 154 (2008), no. 1, 39–49.
[9] L.Caffarelli, P.F.Guan and X.N.Ma, A constant rank theorem for solutions of fullynonlinear elliptic equations, Comm. Pure Appl. Math. Vol.60 (2007), 1769–1791.
[10] A.Chang, X.N.Ma and P.Yang, Principal curvature estimates for the convex levelsets of semilinear elliptic equations, Discrete Contin. Dynam. Systems, Vol. 28,No.3 (2010), 1151–1164.
[11] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc.,Englewood Cliffs, N.J., 1976.
[12] J. Dolbeault and R. Monneau, Convexity estimates for nonlinear elliptic equa-tions and application to free boundary problems, Ann. Inst. H. PoincarWAnal. NonLinWaire, 19, (2002), no. 6, 903–926.
[13] R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonicfunctions, J. London Math.Soc. 32, (1957), 286–294.
[14] J.J.Gergen, Note on the Green function of a star-shaped three dimensional region,Amer. J. Math., 53 (1931), 746–752.
[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of SecondOrder, Springer-Verlag, Berlin, 2nd ed., 1998.
[16] S. Gleason and T. Wolff, Lewy’s harmonic gradient maps in higher dimensions.Comm. Partial Diff. Equations, 16, (1991), 1925–1968.
[17] P.F.Guan, C.S. Lin and X.N. Ma, The Christoffel-Minkowski problem II:WeingartenCurvature Equations, Chinese Ann. Math. Ser. B 27 (2006), no. 6, 595–614.
[18] P.F. Guan, C.S. Lin and G.F. Wang, Local gradient estimates for quotient equationsin conformal geometry, Internat. J. Math. 18 (2007), no. 4, 349–361.
[19] P.F. Guan and X.N.Ma, Christoffel-Minkowski problem I:Convexity of solutions ofa Hessian equations , Invent.Math.151(2003),553–577.
[20] P.F. Guan and X.N.Ma, Convex solutions of fully nonlinear elliptic equations inclassical differential geometry , Contemp. Math., 367,(2005),115–127.
[21] P.F. Guan, X.N.Ma and F.Zhou, Christoffel-Minkowski problem III: Existence andConvexity of admissible solutions , Comm. Pure Appl. Math. 59 (2006), no. 9,1352–1376.
[22] P.F.Guan and L.Xu, Convexity estimates for level sets of quasiconcave solutions tofully nonlinear elliptic equations, Arxiv:1004.1187v1.
[23] P.F. Guan and X.J. Wang, On a Monge-Amp`ere equation arising in geometricoptics, J. Differential Geom. 48 (1998), no. 2, 205–223.
[24] P.F.Guan and G.F.Wang, Local estimates for a class of fully nonlinear equationsarising from conformal geometry. Int. Math. Res. Not. 2003, no. 26, 1413–1432.
[25] C.Q.Hu, X.N. Ma and C.L. Shen, On the Christoffel-Minkowski problem of Fiery’sP-sum, Calc. Var. Partial Differential Equations 21 (2004), no. 2, 137–155.
[26] W.H. Huang, Superharmonicity of curvatures for surfaces of constant mean curva-ture, Pacific J. Math., 152, (1992), no.2, 291–318.
[27] K.Huang, W.Zhang, Convexity of the smallest principal curvature of the con-vex level sets of some quasi-linear elliptic equations with respect to the height ,arXiv:1005.1515v1.
[28] G.Huisken, C.Sinestrari, Convexity estimates for mean curvature flow and singu-larities of mean convex surfaces,Acta Math. 183 (1999), no. 1, 45–70.
[29] J. Jost, X. N. Ma and Q.Z. Ou, Curvature estimates in dimensions 2 and 3 for thelevel sets of p-harmonic functions in convex rings, to appear in Trans. Amer. Math.Soc..
[30] B. Kawohl, Rearrangements and convexity of level sets in PDE, Lectures Notes inMath., 1150, Springer-Verlag, Berlin, 1985.
[31] N. Korevaar and J. Lewis, Convex solutions of certain elliptic equations have con-stant rank Hessians, Arch. Rational Mech. Anal. 97 (1987), no. 1, 19–32.
[32] N. J.Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm.Partial Differ. Equations, 15(4), 1990, 541–556.
[33] J.L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66,(1977), 201–224.
[34] H. Lewy, On the non-vanishing of the Jacobian of a homeomorphism by harmonicGradients, Annals of Math., 88, (1968), 518–529.
[35] Y.Y. Li, Degenerate conformally invariant fully nonlinear elliptic equations , Arch.Ration. Mech. Anal. 186 (2007), no. 1, 25–51.
[36] P.Liu, X.N.Ma and L.Xu, A Brunn-Minkowski inequatlity for the Hessian eigenvaluein three-dimensional convex domain, Adv.Math., doi:10.1016/j.aim.2010.04.003.
[37] M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll.Un. Mat. Ital. A 6, (1983), 71–75.
[38] M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes,J. Differential Equations, 67, (1987), 344–358.
[39] M. Longinetti and P.Salani, On the Hessian matrix and Minkowski addition ofquasiconvex functions, J. Math. Pures Appl., 88, (2007), 276–292.
[40] X. N. Ma and Q.Z. Ou, The convexity of level sets for solutions to partial differentialequations, Trends in Partial Differential Equations, ALM 10, 295–322, 2009.
[41] X.N.Ma, Q.Z. Ou and W.Zhang, Gaussian Curvature estimates for the convex levelsets of ??-harmonic functions, Comm. Pure Appl.,Vol. 63,(2010), 0935–0971.
[42] X.N.Ma and W.Zhang, The concavity of the Gaussian curvature of the convex levelsets of p-harmonic functions with respect to the height, preprint.
[43] V.I.Oliker, Kummer configurations and ????-reflector problems, Results in Math.,56(2009), 519–535.
[44] M.Ortel and W.Schneider, Curvature of level curves of harmonic functions,Cand.Math.Bull.26, (1983), no.4, 399–405.
[45] J.P. Rosay and W. Rudin, A maximum principle for sums of subharmonic func-tions,and the convexity of level sets, Michigan Math. J., 36, (1989), 95–111.
[46] M. Shiffman, On surfaces of stationary area bounded by two circles, or convexcurves, in parallel planes, Annals of Math., 63, (1956), 77–90.
[47] S.-Y.S. Chen, Local estimates for some fully nonlinear elliptic equations, Int. Math.Res. Not.,(2005), no. 55, 3403–3425.
[48] G. Talenti, On functions, whose lines of steepest descent bend proportionally to levellines, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, (1983), no. 4, 587–605.
[49] X.J.Wang, On the design of reflector antenna, Inverse Problems, 12 (1996), no. 3,351–375.
[50] X.J.Wang, A Priori estimates and existence for a class of fully nonlinear ellipticequations in conformal geometry, Chinese Ann. Math. Ser. B 27 (2006), no. 2,169–178.
[51] P.H. Wang and W.Zhang, Gaussian curvature estimates for the convex level setsfor some nonlinear partial differential equations, ArXiv:1003.2057v1.
[52] S.T.Yau, Harmonic functions on complete Riemannian manifolds, Comm.PureAppl. Math., 28(1975), 201–228.
[53] L.Xu, A microscopic convexity theorem of level sets for solutions to elliptic equa-tions, to appear in Calc. Var. Partial Differential Equations.
[2] B.J. Bian, P. Guan, X.N. Ma and L. Xu, A microscopic convexity principle for thelevel sets of solution for nonlinear elliptic partial differential equations, to appearin Indiana Univ. Math. J..
[3] B.J.Bian and P.F.Guan, A microscopic convexity principle for nonlinear differentialequations, Invent. Math. 177 (2009), no. 2, 307–335.
[4] C. Bianchini, M. Longinetti and P. Salani, Quasiconcave solutions to elliptic prob-lems in convex rings, Indiana Univ. Math. J. 58 (2009), no. 4, 1565–1589.
[5] L.A. Caffarelli and A. Friedman, Convexity of solutions of some semilinear ellipticequations, Duke Math. J., 52, (1985), 431–455.
[6] L.A. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equa-tions IV: Starshaped compact Weigarten hypersurfaces, Current topics in partialdifferential equations, 1–26, Kinokuniya, Tokyo, 1986.
[7] L.A. Caffarelli and J. Spruck, Convexity properties of solutions to some classicalvariational problems, Comm. Partial Differ. Equations, 7, (1982), 1337–1379.
[8] L.A.Caffarelli and V.I.Oliker, Weak solutions of one inverse problem in geoetricoptics, J. Math. Sci. (N. Y.) 154 (2008), no. 1, 39–49.
[9] L.Caffarelli, P.F.Guan and X.N.Ma, A constant rank theorem for solutions of fullynonlinear elliptic equations, Comm. Pure Appl. Math. Vol.60 (2007), 1769–1791.
[10] A.Chang, X.N.Ma and P.Yang, Principal curvature estimates for the convex levelsets of semilinear elliptic equations, Discrete Contin. Dynam. Systems, Vol. 28,No.3 (2010), 1151–1164.
[11] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc.,Englewood Cliffs, N.J., 1976.
[12] J. Dolbeault and R. Monneau, Convexity estimates for nonlinear elliptic equa-tions and application to free boundary problems, Ann. Inst. H. PoincarWAnal. NonLinWaire, 19, (2002), no. 6, 903–926.
[13] R. Gabriel, A result concerning convex level surfaces of 3-dimensional harmonicfunctions, J. London Math.Soc. 32, (1957), 286–294.
[14] J.J.Gergen, Note on the Green function of a star-shaped three dimensional region,Amer. J. Math., 53 (1931), 746–752.
[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of SecondOrder, Springer-Verlag, Berlin, 2nd ed., 1998.
[16] S. Gleason and T. Wolff, Lewy’s harmonic gradient maps in higher dimensions.Comm. Partial Diff. Equations, 16, (1991), 1925–1968.
[17] P.F.Guan, C.S. Lin and X.N. Ma, The Christoffel-Minkowski problem II:WeingartenCurvature Equations, Chinese Ann. Math. Ser. B 27 (2006), no. 6, 595–614.
[18] P.F. Guan, C.S. Lin and G.F. Wang, Local gradient estimates for quotient equationsin conformal geometry, Internat. J. Math. 18 (2007), no. 4, 349–361.
[19] P.F. Guan and X.N.Ma, Christoffel-Minkowski problem I:Convexity of solutions ofa Hessian equations , Invent.Math.151(2003),553–577.
[20] P.F. Guan and X.N.Ma, Convex solutions of fully nonlinear elliptic equations inclassical differential geometry , Contemp. Math., 367,(2005),115–127.
[21] P.F. Guan, X.N.Ma and F.Zhou, Christoffel-Minkowski problem III: Existence andConvexity of admissible solutions , Comm. Pure Appl. Math. 59 (2006), no. 9,1352–1376.
[22] P.F.Guan and L.Xu, Convexity estimates for level sets of quasiconcave solutions tofully nonlinear elliptic equations, Arxiv:1004.1187v1.
[23] P.F. Guan and X.J. Wang, On a Monge-Amp`ere equation arising in geometricoptics, J. Differential Geom. 48 (1998), no. 2, 205–223.
[24] P.F.Guan and G.F.Wang, Local estimates for a class of fully nonlinear equationsarising from conformal geometry. Int. Math. Res. Not. 2003, no. 26, 1413–1432.
[25] C.Q.Hu, X.N. Ma and C.L. Shen, On the Christoffel-Minkowski problem of Fiery’sP-sum, Calc. Var. Partial Differential Equations 21 (2004), no. 2, 137–155.
[26] W.H. Huang, Superharmonicity of curvatures for surfaces of constant mean curva-ture, Pacific J. Math., 152, (1992), no.2, 291–318.
[27] K.Huang, W.Zhang, Convexity of the smallest principal curvature of the con-vex level sets of some quasi-linear elliptic equations with respect to the height ,arXiv:1005.1515v1.
[28] G.Huisken, C.Sinestrari, Convexity estimates for mean curvature flow and singu-larities of mean convex surfaces,Acta Math. 183 (1999), no. 1, 45–70.
[29] J. Jost, X. N. Ma and Q.Z. Ou, Curvature estimates in dimensions 2 and 3 for thelevel sets of p-harmonic functions in convex rings, to appear in Trans. Amer. Math.Soc..
[30] B. Kawohl, Rearrangements and convexity of level sets in PDE, Lectures Notes inMath., 1150, Springer-Verlag, Berlin, 1985.
[31] N. Korevaar and J. Lewis, Convex solutions of certain elliptic equations have con-stant rank Hessians, Arch. Rational Mech. Anal. 97 (1987), no. 1, 19–32.
[32] N. J.Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm.Partial Differ. Equations, 15(4), 1990, 541–556.
[33] J.L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66,(1977), 201–224.
[34] H. Lewy, On the non-vanishing of the Jacobian of a homeomorphism by harmonicGradients, Annals of Math., 88, (1968), 518–529.
[35] Y.Y. Li, Degenerate conformally invariant fully nonlinear elliptic equations , Arch.Ration. Mech. Anal. 186 (2007), no. 1, 25–51.
[36] P.Liu, X.N.Ma and L.Xu, A Brunn-Minkowski inequatlity for the Hessian eigenvaluein three-dimensional convex domain, Adv.Math., doi:10.1016/j.aim.2010.04.003.
[37] M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll.Un. Mat. Ital. A 6, (1983), 71–75.
[38] M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes,J. Differential Equations, 67, (1987), 344–358.
[39] M. Longinetti and P.Salani, On the Hessian matrix and Minkowski addition ofquasiconvex functions, J. Math. Pures Appl., 88, (2007), 276–292.
[40] X. N. Ma and Q.Z. Ou, The convexity of level sets for solutions to partial differentialequations, Trends in Partial Differential Equations, ALM 10, 295–322, 2009.
[41] X.N.Ma, Q.Z. Ou and W.Zhang, Gaussian Curvature estimates for the convex levelsets of ??-harmonic functions, Comm. Pure Appl.,Vol. 63,(2010), 0935–0971.
[42] X.N.Ma and W.Zhang, The concavity of the Gaussian curvature of the convex levelsets of p-harmonic functions with respect to the height, preprint.
[43] V.I.Oliker, Kummer configurations and ????-reflector problems, Results in Math.,56(2009), 519–535.
[44] M.Ortel and W.Schneider, Curvature of level curves of harmonic functions,Cand.Math.Bull.26, (1983), no.4, 399–405.
[45] J.P. Rosay and W. Rudin, A maximum principle for sums of subharmonic func-tions,and the convexity of level sets, Michigan Math. J., 36, (1989), 95–111.
[46] M. Shiffman, On surfaces of stationary area bounded by two circles, or convexcurves, in parallel planes, Annals of Math., 63, (1956), 77–90.
[47] S.-Y.S. Chen, Local estimates for some fully nonlinear elliptic equations, Int. Math.Res. Not.,(2005), no. 55, 3403–3425.
[48] G. Talenti, On functions, whose lines of steepest descent bend proportionally to levellines, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, (1983), no. 4, 587–605.
[49] X.J.Wang, On the design of reflector antenna, Inverse Problems, 12 (1996), no. 3,351–375.
[50] X.J.Wang, A Priori estimates and existence for a class of fully nonlinear ellipticequations in conformal geometry, Chinese Ann. Math. Ser. B 27 (2006), no. 2,169–178.
[51] P.H. Wang and W.Zhang, Gaussian curvature estimates for the convex level setsfor some nonlinear partial differential equations, ArXiv:1003.2057v1.
[52] S.T.Yau, Harmonic functions on complete Riemannian manifolds, Comm.PureAppl. Math., 28(1975), 201–228.
[53] L.Xu, A microscopic convexity theorem of level sets for solutions to elliptic equa-tions, to appear in Calc. Var. Partial Differential Equations.