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A gradient estimate for harmonic functions sharing the same zeros

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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 $\log |u/v|$.
    Mathematics Subject Classification: 31B05, 35J15.

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  • [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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