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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

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.  Google Scholar

[2]

L. Caffarelli, The obstacle problem revisited,, J. Fourier Anal. Appl., 4 (1998), 383.  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.  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, (2014).   Google Scholar

[5]

F. Ferrari, On boundary behavior of harmonic functions in Hölder domains,, J. Fourier Anal. Appl., 4 (1998), 447.  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.  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.  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.  doi: 10.1007/BF02790009.  Google Scholar

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.  doi: 10.1215/S0012-7094-91-06408-2.  Google Scholar

[2]

L. Caffarelli, The obstacle problem revisited,, J. Fourier Anal. Appl., 4 (1998), 383.  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.  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, (2014).   Google Scholar

[5]

F. Ferrari, On boundary behavior of harmonic functions in Hölder domains,, J. Fourier Anal. Appl., 4 (1998), 447.  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.  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.  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.  doi: 10.1007/BF02790009.  Google Scholar

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