一类椭圆方程正解的水平集凸性的曲率估计
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摘要
本文共分三节.
第一节为本文的引言,同时给出了主要定理：设Ω是R2中的一有界光滑区域,u∈C4(Ω)∩C2(Ω),且u是椭圆方程Δu=u+u-1|▽u|2
在Ω中的一个正解,若u的水平集对于外法向量▽u是严格凸的,则函数u-2|▽u|2k在边界可以取到极小值.其中,K是u的水平集的曲率.
第二节中的预备知识我们分成了两部分.第一部分介绍了极大值原理的内容以及证明；接下来的第二部分简单叙述了微分几何学中图的凸性定义和函数水平集凸性的概念,给出了函数水平集的曲率矩阵.
第三节为主要定理的证明,运用先验估计的思想,通过一系列的求导计算分析,根据第二节中的极大值原理使定理得以证明.
This thesis consists of three sections.
The first section is the introduction, and we give the main theorem:
LetΩbe a smooth bounded domain in R2 and u∈C4(Ω)∩C2(Q) be a positive solution of the elliptic equation inΩ, i.e.Δu=u+u-1|▽u|2
the level sets of u are strictly convex with respect to normal▽u. Then the function u-2|▽u|2k attains its minimum on the boundary. Where k is the curvature of the level sets of u.
In the second section, we divide it into two parts. Firstly, we introduce the content and proof about maximum principle. Later, we give the convexity of the graph in differential geom-etry; then introduce brief definitions on the convexity of the level sets and obtain the curvature matrix of the level sets of a function.
The third section is the proof of the main theorem. We apply the idea of the priori estimates, by some formal computations for the derivative terms. With the maximum principle, we attain the proof of the main theorem at last.
第一节为本文的引言,同时给出了主要定理：设Ω是R2中的一有界光滑区域,u∈C4(Ω)∩C2(Ω),且u是椭圆方程Δu=u+u-1|▽u|2
在Ω中的一个正解,若u的水平集对于外法向量▽u是严格凸的,则函数u-2|▽u|2k在边界可以取到极小值.其中,K是u的水平集的曲率.
第二节中的预备知识我们分成了两部分.第一部分介绍了极大值原理的内容以及证明；接下来的第二部分简单叙述了微分几何学中图的凸性定义和函数水平集凸性的概念,给出了函数水平集的曲率矩阵.
第三节为主要定理的证明,运用先验估计的思想,通过一系列的求导计算分析,根据第二节中的极大值原理使定理得以证明.
This thesis consists of three sections.
The first section is the introduction, and we give the main theorem:
LetΩbe a smooth bounded domain in R2 and u∈C4(Ω)∩C2(Q) be a positive solution of the elliptic equation inΩ, i.e.Δu=u+u-1|▽u|2
the level sets of u are strictly convex with respect to normal▽u. Then the function u-2|▽u|2k attains its minimum on the boundary. Where k is the curvature of the level sets of u.
In the second section, we divide it into two parts. Firstly, we introduce the content and proof about maximum principle. Later, we give the convexity of the graph in differential geom-etry; then introduce brief definitions on the convexity of the level sets and obtain the curvature matrix of the level sets of a function.
The third section is the proof of the main theorem. We apply the idea of the priori estimates, by some formal computations for the derivative terms. With the maximum principle, we attain the proof of the main theorem at last.
引文
[1]F. H. Lin, Q. Han. Elliptic partial differential equations, [M]. New York:Courant Institute of Mathematical Sciences,2000.
[2]X. N. Ma, Qianzhong Ou, Wei. Zhang. Gaussian curvature estimates for the convex level sets of p-harmonic functions, [J]. Comm. Pure Appl. Math.,2010,63(7):935-971.
[3]R. Gabriel. A result concerning convex level surfaces of 3-dimensional harmonic functions, [J]. London Math.Soc.,1957,32:286-294.
[4]J. L. Lewis. Capacitary functions in convex rings, [J]. Arch. Rational Mech. Anal.,1977,201-224.
[5]L. Caffarelli, A. Friedman. Convexity of solutions of some semilinear elliptic equations, [J]. Duke Math.,1985,52:431-455.
[6]B. J. Bian, P. Guan, X. N. Ma, L. Xu. A microscopic convexity principle for the level sets of solution for nonlinear elliptic partial differential equations, preprint.
[7]C. Bianchini, M. Longinetti, P. Salani. Quasiconcave solutions to elliptic problems in convex rings, [J]. Indiana Univ. Math. Journal.,2009,58:1565-1590.
[8]X. N. Ma, Qianzhong Ou. The convexity of level sets for solutions to partial differential equations, preprint.
[9]J. Jost, X. N. Ma, Qianzhong Ou. Curvature estimates in dimensions 2 and 3 for the level sets of p-harmonic functions in convex rings, preprint.
[10]L. Caffarelli, L. Nirenberg, J. Spruck. Nonlinear second order elliptic equations Ⅳ:Starshaped compact Weigarten hypersurfaces, Current topics in partial differential equations, Current topics in partial differential equations, Y.Ohya, K.Kasahara and N.Shimakura (eds), [M]. Kinokunize, Tokyo.,1985,1-26.
[11]L. Caffarelli, J. Spruck. Convexity properties of solutions to some classical variational problems, [J]. Comm. Partial Differ. Equations.,1982,7:1337-1379.
[12]N. Korevaar. Convexity of level sets for solutions to elliptic ring problems, [J]. Comm. Partial Differ. Equations.,1990,15(4):541-556.
[13]P. H. Wang, Wei. Zhang. Gaussian curvature estimates for the convex level sets of solutions for some nonlinear elliptic partial differential equations, preprint. 参考文献
[14]G. Chiti, M. Longinetti. Differential inequalities for Minkowski funtionals of level sets, [J]. Inter-nat. Ser. Numer. Math.103 (1992),109-127.
[15]R. Schneider. Convex bodies:The Brunn-Minkowski theory, [M]. Cambridge University.,1993.
[16]B. Kawohl. Rearrangements and convexity of level sets in PDE, [R]. Springer-Verlag, Berlin,1985, 1150.
[17]M. Spivak. A comprehensive introduction to differential geometry, [M]. Vol. Ⅳ. Second edition. Publish or Perish, Inc.,1979.
[18]M. Shiffman. On surfaces of stationary area bounded by two circles, or convex curves in parallel planes, [J]. Annals of Math.,1956,63:77-90.
[19]M. Longinetti. Convexity of the level lines of harmonic functions, [J]. (Italian) Boll. Un. Mat. Ital., 1983, A 6:71-75.
[20]M. Longinetti. On minimal surfaces bounded by two convex curves in parallel planes, [J]. Diff. Equations.,1987,67:344-358.
[21]X. N. Ma, Jiang Ye, Yunhua Ye. Principal curvature estimates for the level sets of harmonic func-tions and minimal graph in R3, preprint.
[22]M. Longinetti, P. Salani. On the Hessian matrix and Minkowski addition of quasiconvex functions, [J]. Math. Pures Appl.,2007,88(9):276-292.
[23]G. Talenti. On functions, whose lines of steepest descent bend proportionally to level lines, [J]. Ann. Scuola Norm. Sup. Pisa Cl. Sci.,1983,10(4):587-605.
[24]X. N. Ma, W. Zhang. The concavity of the Gaussian curvature of the convex level sets of p-harmonic funtion with respect to the height, preprint.
[25]M. P. do Carmo. Differential geometry of curves and surfaces, [M]. Prentice-Hall, Inc., Englewood Cliffs, N.J.,1976.
[26]J. Dolbeault and R. Monneau. Convexity estimates for nonlinear elliptic equtions and application to free boundary problems, [J]. Ann. Inst. H. Poincar Anal. Non Linaire, (2002), no.6,903-926.
[27]J. Rosay and W. Rudin. A maximum principle for sums of subharmonic funtions, and the convexity of level sets, [J]. Michigan Math. J., (1989),36:95-111.
[28]M. Ortel, W. Schneider. Curvature of level curves of harmonic funtions, [J]. Canad. Math. Bull. (1983),26(4),399-405.
[29]奥列尼克,郭思旭.偏微分方程讲义[M].北京：高等教育出版社,2008.
[30]陈维桓等.微分几何[M].北京：北京大学出版社,2006.
[2]X. N. Ma, Qianzhong Ou, Wei. Zhang. Gaussian curvature estimates for the convex level sets of p-harmonic functions, [J]. Comm. Pure Appl. Math.,2010,63(7):935-971.
[3]R. Gabriel. A result concerning convex level surfaces of 3-dimensional harmonic functions, [J]. London Math.Soc.,1957,32:286-294.
[4]J. L. Lewis. Capacitary functions in convex rings, [J]. Arch. Rational Mech. Anal.,1977,201-224.
[5]L. Caffarelli, A. Friedman. Convexity of solutions of some semilinear elliptic equations, [J]. Duke Math.,1985,52:431-455.
[6]B. J. Bian, P. Guan, X. N. Ma, L. Xu. A microscopic convexity principle for the level sets of solution for nonlinear elliptic partial differential equations, preprint.
[7]C. Bianchini, M. Longinetti, P. Salani. Quasiconcave solutions to elliptic problems in convex rings, [J]. Indiana Univ. Math. Journal.,2009,58:1565-1590.
[8]X. N. Ma, Qianzhong Ou. The convexity of level sets for solutions to partial differential equations, preprint.
[9]J. Jost, X. N. Ma, Qianzhong Ou. Curvature estimates in dimensions 2 and 3 for the level sets of p-harmonic functions in convex rings, preprint.
[10]L. Caffarelli, L. Nirenberg, J. Spruck. Nonlinear second order elliptic equations Ⅳ:Starshaped compact Weigarten hypersurfaces, Current topics in partial differential equations, Current topics in partial differential equations, Y.Ohya, K.Kasahara and N.Shimakura (eds), [M]. Kinokunize, Tokyo.,1985,1-26.
[11]L. Caffarelli, J. Spruck. Convexity properties of solutions to some classical variational problems, [J]. Comm. Partial Differ. Equations.,1982,7:1337-1379.
[12]N. Korevaar. Convexity of level sets for solutions to elliptic ring problems, [J]. Comm. Partial Differ. Equations.,1990,15(4):541-556.
[13]P. H. Wang, Wei. Zhang. Gaussian curvature estimates for the convex level sets of solutions for some nonlinear elliptic partial differential equations, preprint. 参考文献
[14]G. Chiti, M. Longinetti. Differential inequalities for Minkowski funtionals of level sets, [J]. Inter-nat. Ser. Numer. Math.103 (1992),109-127.
[15]R. Schneider. Convex bodies:The Brunn-Minkowski theory, [M]. Cambridge University.,1993.
[16]B. Kawohl. Rearrangements and convexity of level sets in PDE, [R]. Springer-Verlag, Berlin,1985, 1150.
[17]M. Spivak. A comprehensive introduction to differential geometry, [M]. Vol. Ⅳ. Second edition. Publish or Perish, Inc.,1979.
[18]M. Shiffman. On surfaces of stationary area bounded by two circles, or convex curves in parallel planes, [J]. Annals of Math.,1956,63:77-90.
[19]M. Longinetti. Convexity of the level lines of harmonic functions, [J]. (Italian) Boll. Un. Mat. Ital., 1983, A 6:71-75.
[20]M. Longinetti. On minimal surfaces bounded by two convex curves in parallel planes, [J]. Diff. Equations.,1987,67:344-358.
[21]X. N. Ma, Jiang Ye, Yunhua Ye. Principal curvature estimates for the level sets of harmonic func-tions and minimal graph in R3, preprint.
[22]M. Longinetti, P. Salani. On the Hessian matrix and Minkowski addition of quasiconvex functions, [J]. Math. Pures Appl.,2007,88(9):276-292.
[23]G. Talenti. On functions, whose lines of steepest descent bend proportionally to level lines, [J]. Ann. Scuola Norm. Sup. Pisa Cl. Sci.,1983,10(4):587-605.
[24]X. N. Ma, W. Zhang. The concavity of the Gaussian curvature of the convex level sets of p-harmonic funtion with respect to the height, preprint.
[25]M. P. do Carmo. Differential geometry of curves and surfaces, [M]. Prentice-Hall, Inc., Englewood Cliffs, N.J.,1976.
[26]J. Dolbeault and R. Monneau. Convexity estimates for nonlinear elliptic equtions and application to free boundary problems, [J]. Ann. Inst. H. Poincar Anal. Non Linaire, (2002), no.6,903-926.
[27]J. Rosay and W. Rudin. A maximum principle for sums of subharmonic funtions, and the convexity of level sets, [J]. Michigan Math. J., (1989),36:95-111.
[28]M. Ortel, W. Schneider. Curvature of level curves of harmonic funtions, [J]. Canad. Math. Bull. (1983),26(4),399-405.
[29]奥列尼克,郭思旭.偏微分方程讲义[M].北京：高等教育出版社,2008.
[30]陈维桓等.微分几何[M].北京：北京大学出版社,2006.