A note on higher regularity boundary Harnack inequality

Daniela De Silva; Ovidiu  Savin

doi:10.3934/dcds.2015.35.6155

Article Contents

2015, Volume 35, Issue 12: 6155-6163. Doi: 10.3934/dcds.2015.35.6155

This issue Previous Article Complexity and regularity of maximal energy domains for the wave equation with fixed initial data Next Article The Hessian Sobolev inequality and its extensions

A note on higher regularity boundary Harnack inequality

Daniela De Silva^1, and
Ovidiu Savin^2,

1.
Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027
2.
Department of Mathematics, Columbia University, New York, NY 10027

Received: February 28, 2014

Published: November 2015

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Abstract

We show that the quotient of a harmonic function and a positive harmonic function, both vanishing on the boundary of a $C^{k,\alpha}$ domain is of class $C^{k,\alpha}$ up to the boundary.

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Mathematics Subject Classification: 35B65, 35R35.

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References

[1]	R. Bañuelos, R. F. Bass and K. Burdzy, Hölder domains and the boundary Harnack principle, Duke Math. J., 64 (1991), 195-200.doi: 10.1215/S0012-7094-91-06408-2.
[2]	L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402.doi: 10.1007/BF02498216.
[3]	L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of non-negative solutions of elliptic operators in divergence form, Indiana Math. J., 30 (1981), 621-640.doi: 10.1512/iumj.1981.30.30049.
[4]	D. De Silva and O. Savin, $C^\infty$ regularity of certain thin free boundaries, submitted, arXiv:1402.1098, 2014.
[5]	F. Ferrari, On boundary behavior of harmonic functions in Hölder domains, J. Fourier Anal. Appl., 4 (1998), 447-461.doi: 10.1007/BF02498219.
[6]	R. A. Hunt and R. L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc., 132 (1968), 307-322.doi: 10.1090/S0002-9947-1968-0226044-7.
[7]	D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math., 46 (1982), 80-147.doi: 10.1016/0001-8708(82)90055-X.
[8]	D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free boundary problems, J. Analyse Math., 34 (1978), 86-119 (1979).doi: 10.1007/BF02790009.