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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

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 $\log |u/v|$.
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,, \emph{Ann. Inst. Fourier (Grenoble)}, 28 (1978), 169. doi: 10.5802/aif.720.

[2]

Z. Balogh and A. Volberg, Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications,, \emph{Rev. Mat. Iberoamericana}, 12 (1996), 299. 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,, \emph{Indiana Univ. Math. J.}, 30 (1981), 621. 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, (1998).

[5]

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

[6]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, (Russian) \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, 47 (1983), 75.

[7]

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

[8]

A. Logunov and E. Malinnikova, On ratios of harmonic functions,, preprint, (2014).

[9]

N. Nadirashvili, Harmonic functions with bounded number of nodal domains,, \emph{Appl. Anal.}, 71 (1999), 187. doi: 10.1080/00036819908840712.

[10]

I. Popovici and A. Volberg, Boundary Harnack principle for Denjoy domains,, \emph{Complex Variables Theory Appl.}, 37 (1998), 471. doi: 10.1080/17476939808815145.

[11]

L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations,, preprint, (2013).

[12]

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

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,, \emph{Ann. Inst. Fourier (Grenoble)}, 28 (1978), 169. doi: 10.5802/aif.720.

[2]

Z. Balogh and A. Volberg, Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications,, \emph{Rev. Mat. Iberoamericana}, 12 (1996), 299. 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,, \emph{Indiana Univ. Math. J.}, 30 (1981), 621. 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, (1998).

[5]

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

[6]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, (Russian) \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, 47 (1983), 75.

[7]

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

[8]

A. Logunov and E. Malinnikova, On ratios of harmonic functions,, preprint, (2014).

[9]

N. Nadirashvili, Harmonic functions with bounded number of nodal domains,, \emph{Appl. Anal.}, 71 (1999), 187. doi: 10.1080/00036819908840712.

[10]

I. Popovici and A. Volberg, Boundary Harnack principle for Denjoy domains,, \emph{Complex Variables Theory Appl.}, 37 (1998), 471. doi: 10.1080/17476939808815145.

[11]

L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations,, preprint, (2013).

[12]

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

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