American Institute of Mathematical Sciences

Advanced
  • Previous Article
    From local to global equilibrium states: Thermodynamic formalism via an inducing scheme
  • ERA-MS Home
  • This Volume
  • Next Article
    Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
  2014, 21: 62-71. doi: 10.3934/era.2014.21.62

A gradient estimate for harmonic functions sharing the same zeros

Dan Mangoubi 1,

1. 

Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904

Received  June 2013 Revised  December 2013 Published  May 2014

Let $u, v$ be two harmonic functions in $\{|z|<2\}\subset\mathbb{C}$ which have exactly the same set $Z$ of zeros. We observe that $\big|\nabla\log |u/v|\big|$ is bounded in the unit disk by a constant which depends on $Z$ only. In case $Z=\emptyset$ this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives only Hölder estimates on $\log |u/v|$.
Keywords: gradient estimates, nodal set, Harnack, Li-Yau, harmonic functions, boundary Harnack principle..
Mathematics Subject Classification: 31B05, 35J1.
Citation: 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
References:
[1]

A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble), 28 (1978), 169-213. doi: 10.5802/aif.720.  Google Scholar

[2]

Z. Balogh and A. Volberg, Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications, Rev. Mat. Iberoamericana, 12 (1996), 299-336. doi: 10.4171/RMI/200.  Google Scholar

[3]

L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981), 621-640. doi: 10.1512/iumj.1981.30.30049.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[5]

D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46 (1982), 80-147. doi: 10.1016/0001-8708(82)90055-X.  Google Scholar

[6]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108; translation in Math. USSR-Izv., 22 (1984), 67-98.  Google Scholar

[7]

P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. doi: 10.1007/BF02399203.  Google Scholar

[8]

A. Logunov and E. Malinnikova, On ratios of harmonic functions, preprint, arXiv:1402.2888, 2014. Google Scholar

[9]

N. Nadirashvili, Harmonic functions with bounded number of nodal domains, Appl. Anal., 71 (1999), 187-196. doi: 10.1080/00036819908840712.  Google Scholar

[10]

I. Popovici and A. Volberg, Boundary Harnack principle for Denjoy domains, Complex Variables Theory Appl., 37 (1998), 471-490. doi: 10.1080/17476939808815145.  Google Scholar

[11]

L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, preprint, arXiv:1306.6672, 2013. Google Scholar

[12]

J. M. G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble), 28 (1978), 147-167. doi: 10.5802/aif.719.  Google Scholar

show all references

References:
[1]

A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble), 28 (1978), 169-213. doi: 10.5802/aif.720.  Google Scholar

[2]

Z. Balogh and A. Volberg, Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications, Rev. Mat. Iberoamericana, 12 (1996), 299-336. doi: 10.4171/RMI/200.  Google Scholar

[3]

L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981), 621-640. doi: 10.1512/iumj.1981.30.30049.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[5]

D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46 (1982), 80-147. doi: 10.1016/0001-8708(82)90055-X.  Google Scholar

[6]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108; translation in Math. USSR-Izv., 22 (1984), 67-98.  Google Scholar

[7]

P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. doi: 10.1007/BF02399203.  Google Scholar

[8]

A. Logunov and E. Malinnikova, On ratios of harmonic functions, preprint, arXiv:1402.2888, 2014. Google Scholar

[9]

N. Nadirashvili, Harmonic functions with bounded number of nodal domains, Appl. Anal., 71 (1999), 187-196. doi: 10.1080/00036819908840712.  Google Scholar

[10]

I. Popovici and A. Volberg, Boundary Harnack principle for Denjoy domains, Complex Variables Theory Appl., 37 (1998), 471-490. doi: 10.1080/17476939808815145.  Google Scholar

[11]

L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, preprint, arXiv:1306.6672, 2013. Google Scholar

[12]

J. M. G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble), 28 (1978), 147-167. doi: 10.5802/aif.719.  Google Scholar

[1]

Fatma Gamze Düzgün, Ugo Gianazza, Vincenzo Vespri. $1$-dimensional Harnack estimates. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 675-685. doi: 10.3934/dcdss.2016021
[2]

Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
[3]

Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155
[4]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425
[5]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793
[6]

Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99.

[7]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[8]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[9]

Matthew B. Rudd, Heather A. Van Dyke. Median values, 1-harmonic functions, and functions of least gradient. Communications on Pure & Applied Analysis, 2013, 12 (2) : 711-719. doi: 10.3934/cpaa.2013.12.711
[10]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[11]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
[12]

J. Földes, Peter Poláčik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 133-157. doi: 10.3934/dcds.2009.25.133
[13]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[14]

Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975
[15]

Simona Fornaro, Maria Sosio, Vincenzo Vespri. Harnack type inequalities for some doubly nonlinear singular parabolic equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5909-5926. doi: 10.3934/dcds.2015.35.5909
[16]

Shiping Cao, Hua Qiu. Boundary value problems for harmonic functions on domains in Sierpinski gaskets. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1147-1179. doi: 10.3934/cpaa.2020054
[17]

René Henrion. Gradient estimates for Gaussian distribution functions: application to probabilistically constrained optimization problems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 655-668. doi: 10.3934/naco.2012.2.655
[18]

Tadeusz Iwaniec, Gaven Martin, Carlo Sbordone. $L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145
[19]

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
[20]

Laura Abatangelo, Susanna Terracini. Harmonic functions in union of chambers. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5609-5629. doi: 10.3934/dcds.2015.35.5609

2020 Impact Factor: 0.929

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences