# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402. doi: 10.1007/BF02498216.  Google Scholar [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.  Google Scholar [4] D. De Silva and O. Savin, $C^\infty$ regularity of certain thin free boundaries, submitted, arXiv:1402.1098, 2014. Google Scholar [5] F. Ferrari, On boundary behavior of harmonic functions in Hölder domains, J. Fourier Anal. Appl., 4 (1998), 447-461. doi: 10.1007/BF02498219.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [2] L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402. doi: 10.1007/BF02498216.  Google Scholar [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.  Google Scholar [4] D. De Silva and O. Savin, $C^\infty$ regularity of certain thin free boundaries, submitted, arXiv:1402.1098, 2014. Google Scholar [5] F. Ferrari, On boundary behavior of harmonic functions in Hölder domains, J. Fourier Anal. Appl., 4 (1998), 447-461. doi: 10.1007/BF02498219.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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