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 & 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, (1973), 5.

[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. 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. 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, (1985), 1.

[6]

L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems,, Comm. Partial Differ. Equations, 7 (1982), 1337. 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), (2010), 1151.

[8]

J. Dolbeault and R. Monneau, Convexity estimates for nonlinear elliptic equations and application to free boundary problems,, Ann. Inst. H. Poincaré Anal. Non Lin$\acutee$aire, 19 (2002), 903. 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. 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. 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. 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 (1150).

[14]

N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems,, Comm. Partial Differ. Equations, 15 (1990), 541. doi: doi:10.1080/03605309908820698.

[15]

J. L. Lewis, Capacitary functions in convex rings,, Arch. Rational Mech. Anal., 66 (1977), 201. 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. doi: doi:10.2307/1970723.

[17]

M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A,, {\bf 6} (1983), 6 (1983), 71.

[18]

M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes,, J. Diff. Equations, 67 (1987), 344. 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., (2010), 935.

[20]

M. Ortel and W. Schneider, Curvature of level curves of harmonic functions,, Canad. Math. Bull., 26 (1983), 399.

[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. 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. 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.

show all references

References:
[1]

L. V. Ahlfors, "Conformal Invariants: Topics in Geometric Function Theory,", McGraw-Hill Series in Higher Mathematics, (1973), 5.

[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. 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. 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, (1985), 1.

[6]

L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems,, Comm. Partial Differ. Equations, 7 (1982), 1337. 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), (2010), 1151.

[8]

J. Dolbeault and R. Monneau, Convexity estimates for nonlinear elliptic equations and application to free boundary problems,, Ann. Inst. H. Poincaré Anal. Non Lin$\acutee$aire, 19 (2002), 903. 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. 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. 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. 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 (1150).

[14]

N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems,, Comm. Partial Differ. Equations, 15 (1990), 541. doi: doi:10.1080/03605309908820698.

[15]

J. L. Lewis, Capacitary functions in convex rings,, Arch. Rational Mech. Anal., 66 (1977), 201. 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. doi: doi:10.2307/1970723.

[17]

M. Longinetti, Convexity of the level lines of harmonic functions, (Italian) Boll. Un. Mat. Ital. A,, {\bf 6} (1983), 6 (1983), 71.

[18]

M. Longinetti, On minimal surfaces bounded by two convex curves in parallel planes,, J. Diff. Equations, 67 (1987), 344. 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., (2010), 935.

[20]

M. Ortel and W. Schneider, Curvature of level curves of harmonic functions,, Canad. Math. Bull., 26 (1983), 399.

[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. 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. 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.

[1]

Sun-Yung Alice Chang, Xi-Nan Ma, Paul Yang. Principal curvature estimates for the convex level sets of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1151-1164. doi: 10.3934/dcds.2010.28.1151

[2]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413

[3]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036

[4]

Dan Mangoubi. A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements, 2014, 21: 62-71. doi: 10.3934/era.2014.21.62

[5]

Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417

[6]

Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301

[7]

Fanghua Lin, Dan Liu. On the Betti numbers of level sets of solutions to elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4517-4529. doi: 10.3934/dcds.2016.36.4517

[8]

Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763

[9]

Ronald A. Knight. Compact minimal sets in continuous recurrent flows. Conference Publications, 1998, 1998 (Special) : 397-407. doi: 10.3934/proc.1998.1998.397

[10]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020

[11]

Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629

[12]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437

[13]

François Hamel, Régis Monneau, Jean-Michel Roquejoffre. Asymptotic properties and classification of bistable fronts with Lipschitz level sets. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 75-92. doi: 10.3934/dcds.2006.14.75

[14]

Pablo G. Barrientos, Abbas Fakhari, Aliasghar Sarizadeh. Density of fiberwise orbits in minimal iterated function systems on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3341-3352. doi: 10.3934/dcds.2014.34.3341

[15]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[16]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231

[17]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[18]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[19]

Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8

[20]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]