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  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

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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

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

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

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

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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

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