A note on higher regularity boundary Harnack inequality

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December 2015, 35(12): 6155-6163. doi: 10.3934/dcds.2015.35.6155

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, United States
2.	Department of Mathematics, Columbia University, New York, NY 10027, United States

Received March 2014 Published May 2015

Abstract
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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.

Keywords: $C^{k, Harnack inequality, obstacle problem., \alpha}$ estimates, boundary regularity, Harmonic functions.

Mathematics Subject Classification: 35B65, 35R3.

Citation: Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155

References:

[1]	R. Bañuelos, R. F. Bass and K. Burdzy, Hölder domains and the boundary Harnack principle,, Duke Math. J., 64 (1991), 195. doi: 10.1215/S0012-7094-91-06408-2.
[2]	L. Caffarelli, The obstacle problem revisited,, J. Fourier Anal. Appl., 4 (1998), 383. 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. doi: 10.1512/iumj.1981.30.30049.
[4]	D. De Silva and O. Savin, $C^\infty$ regularity of certain thin free boundaries,, submitted, (2014).
[5]	F. Ferrari, On boundary behavior of harmonic functions in Hölder domains,, J. Fourier Anal. Appl., 4 (1998), 447. 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. 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. 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. doi: 10.1007/BF02790009.

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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. doi: 10.1215/S0012-7094-91-06408-2.
[2]	L. Caffarelli, The obstacle problem revisited,, J. Fourier Anal. Appl., 4 (1998), 383. 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. doi: 10.1512/iumj.1981.30.30049.
[4]	D. De Silva and O. Savin, $C^\infty$ regularity of certain thin free boundaries,, submitted, (2014).
[5]	F. Ferrari, On boundary behavior of harmonic functions in Hölder domains,, J. Fourier Anal. Appl., 4 (1998), 447. 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. 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. 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. doi: 10.1007/BF02790009.

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