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

A note on higher regularity boundary Harnack inequality

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

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.
Citation: Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete and Continuous Dynamical Systems, 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), 195-200. 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.

show all references

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.

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