A gradient estimate for harmonic functions sharing the same zeros

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	A. Logunov and E. Malinnikova, On ratios of harmonic functions, preprint, arXiv:1402.2888, 2014.
[9]	N. Nadirashvili, Harmonic functions with bounded number of nodal domains, Appl. Anal., 71 (1999), 187-196. doi: 10.1080/00036819908840712.
[10]	I. Popovici and A. Volberg, Boundary Harnack principle for Denjoy domains, Complex Variables Theory Appl., 37 (1998), 471-490. doi: 10.1080/17476939808815145.
[11]	L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, preprint, arXiv:1306.6672, 2013.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	A. Logunov and E. Malinnikova, On ratios of harmonic functions, preprint, arXiv:1402.2888, 2014.
[9]	N. Nadirashvili, Harmonic functions with bounded number of nodal domains, Appl. Anal., 71 (1999), 187-196. doi: 10.1080/00036819908840712.
[10]	I. Popovici and A. Volberg, Boundary Harnack principle for Denjoy domains, Complex Variables Theory Appl., 37 (1998), 471-490. doi: 10.1080/17476939808815145.
[11]	L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, preprint, arXiv:1306.6672, 2013.
[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.

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