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January  2019, 18(1): 107-128. doi: 10.3934/cpaa.2019007

Boundary regularity for a degenerate elliptic equation with mixed boundary conditions

Alassane Niang ,

Department of Mathematics and Computer Sciences, Cheikh Anta Diop University of Dakar (UCAD), B.P. 5005 Dakar-Fann, Senegal

Received  August 2017 Revised  February 2018 Published  August 2018

Fund Project: The author is supported by the NLAGA Project of the Simons foundation and the Post-AIMS bursary of AIMS-SENEGAL

We consider a function $U$ satisfying a degenerate elliptic equation on $\mathbb{R}_ + ^{N + 1}: = (0, +∞)×{\mathbb{R}^N}$ with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain $\Omega\subset{\mathbb{R}^N}$ of class $C^{1, 1}$, whereas the Dirichlet data is on the exterior of $\Omega$. We prove Hölder regularity estimates of $\frac{U}{d_\Omega^s}$, where $d_\Omega$ is a distance function defined as $d_\Omega(z): = \text{dist}(z, {\mathbb{R}^N}\setminus\Omega)$, for $z∈\overline{\mathbb{R}_ + ^{N + 1}}$. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness and blow-up analysis arguments.

Keywords: Degenerate elliptic equation, boundary regularity, fractional Laplacian, blow-up analysis, compactness, Dirichlet-Neumann boundary conditions.
Mathematics Subject Classification: Primary: 35J47, 35J70; Secondary: 35S15.
Citation: Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007
References:
[1]

J. Björn, Regularity at infinity fot a mixed problem for degenerate elliptic operators in a half-cynlider, Math. Scand., 81 (1997), 101-126.  doi: 10.7146/math.scand.a-12868.  Google Scholar

[2]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing, Switzerland, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[3]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[4]

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[5]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[6]

M. Costabel, M. Dauge and R. Duduchava, Asymptotics without logarithmic terms for crack problems, Comm. Partial Differential Equations, 28 (2003), 869-926.  doi: 10.1081/PDE-120021180.  Google Scholar

[7]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[8]

E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble), 32 (1982), 151-182.  doi: 10.5802/aif.883.  Google Scholar

[9]

E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.  Google Scholar

[10]

V. I. Fabrikant, Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Kluwer Academic Publishers, 1991,451 pages.  Google Scholar

[11]

M. M. Fall, Regularity estimates for nonlocal Schrödinger equations, preprint, arXiv: 1711.02206. Google Scholar

[12]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.  Google Scholar

[13]

G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE, 7 (2014), 1649-1682.  doi: 10.2140/apde.2014.7.1649.  Google Scholar

[14]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.  Google Scholar

[15]

G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634.  doi: 10.1016/j.jmaa.2014.07.081.  Google Scholar

[16]

T. Jin, Y. Y. Li and J. Xiong, On a fractional nirenberg problem part i: blow up analysis and compactness solutions, J. Eur. Math. Soc (JEMS), 16 (2014), 1111-1171.  doi: 10.4171/JEMS/456.  Google Scholar

[17]

M. Kassmann and W. R. Madych, Difference quotients and elliptic mixed boundary value problems of second order, Indiana Univ. Math. J., 56 (2007), 1047-1082.  doi: 10.1512/iumj.2007.56.2836.  Google Scholar

[18]

S. Kim and K. Lee, Hölder estimates for singular nonlocal parabolic equations, J. of Funct. Anal., 261 (2011), 3482-3518.  doi: 10.1016/j.jfa.2011.08.010.  Google Scholar

[19]

Serge Levendorskii, Degenerate Elliptic Equations, Springer Netherlands, 1993. doi: 10.1007/978-94-017-1215-6.  Google Scholar

[20]

P. L. Mills and M. P. Dudukovi${\rm{\tilde c}}$, Solution of mixed boundary value problems by integral equations and methods of weighted residuals with application to heat conduction and diffusion-reaction systems, SIAM Journal on Applied Mathematics, 44 (1984), 1076-1091.  doi: 10.1137/0144077.  Google Scholar

[21]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.  doi: 10.2307/1995882.  Google Scholar

[22]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[23]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Duke Mathematical Journal, 165 (2016), 2079-2154.  doi: 10.1215/00127094-3476700.  Google Scholar

[24]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, Journal of Differential Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.  Google Scholar

[25]

G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations, 22 (1997), 869-899.  doi: 10.1080/03605309708821287.  Google Scholar

[26]

J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations, 54 (2015), 615-629.  doi: 10.1007/s00526-014-0798-6.  Google Scholar

[27]

L. Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion, Indiana University Mathematical Journal, 61 (2012), 557-584.  doi: 10.1512/iumj.2012.61.4568.  Google Scholar

[28]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[29]

S. Zaremba, Sur un problème mixte relatif à l'équation de Laplace, (french) [On a mixed problem related to the Laplace equation], Bulletin international de l'Académie des Sciences de Cracovie. Classe des Sciences Mathématiques et Naturelles, Serie A: Sciences mathématiques (French), 313-344. Google Scholar

show all references

References:
[1]

J. Björn, Regularity at infinity fot a mixed problem for degenerate elliptic operators in a half-cynlider, Math. Scand., 81 (1997), 101-126.  doi: 10.7146/math.scand.a-12868.  Google Scholar

[2]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing, Switzerland, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[3]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[4]

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[5]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[6]

M. Costabel, M. Dauge and R. Duduchava, Asymptotics without logarithmic terms for crack problems, Comm. Partial Differential Equations, 28 (2003), 869-926.  doi: 10.1081/PDE-120021180.  Google Scholar

[7]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[8]

E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble), 32 (1982), 151-182.  doi: 10.5802/aif.883.  Google Scholar

[9]

E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.  Google Scholar

[10]

V. I. Fabrikant, Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Kluwer Academic Publishers, 1991,451 pages.  Google Scholar

[11]

M. M. Fall, Regularity estimates for nonlocal Schrödinger equations, preprint, arXiv: 1711.02206. Google Scholar

[12]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.  Google Scholar

[13]

G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE, 7 (2014), 1649-1682.  doi: 10.2140/apde.2014.7.1649.  Google Scholar

[14]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.  Google Scholar

[15]

G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634.  doi: 10.1016/j.jmaa.2014.07.081.  Google Scholar

[16]

T. Jin, Y. Y. Li and J. Xiong, On a fractional nirenberg problem part i: blow up analysis and compactness solutions, J. Eur. Math. Soc (JEMS), 16 (2014), 1111-1171.  doi: 10.4171/JEMS/456.  Google Scholar

[17]

M. Kassmann and W. R. Madych, Difference quotients and elliptic mixed boundary value problems of second order, Indiana Univ. Math. J., 56 (2007), 1047-1082.  doi: 10.1512/iumj.2007.56.2836.  Google Scholar

[18]

S. Kim and K. Lee, Hölder estimates for singular nonlocal parabolic equations, J. of Funct. Anal., 261 (2011), 3482-3518.  doi: 10.1016/j.jfa.2011.08.010.  Google Scholar

[19]

Serge Levendorskii, Degenerate Elliptic Equations, Springer Netherlands, 1993. doi: 10.1007/978-94-017-1215-6.  Google Scholar

[20]

P. L. Mills and M. P. Dudukovi${\rm{\tilde c}}$, Solution of mixed boundary value problems by integral equations and methods of weighted residuals with application to heat conduction and diffusion-reaction systems, SIAM Journal on Applied Mathematics, 44 (1984), 1076-1091.  doi: 10.1137/0144077.  Google Scholar

[21]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.  doi: 10.2307/1995882.  Google Scholar

[22]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[23]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Duke Mathematical Journal, 165 (2016), 2079-2154.  doi: 10.1215/00127094-3476700.  Google Scholar

[24]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, Journal of Differential Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.  Google Scholar

[25]

G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations, 22 (1997), 869-899.  doi: 10.1080/03605309708821287.  Google Scholar

[26]

J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations, 54 (2015), 615-629.  doi: 10.1007/s00526-014-0798-6.  Google Scholar

[27]

L. Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion, Indiana University Mathematical Journal, 61 (2012), 557-584.  doi: 10.1512/iumj.2012.61.4568.  Google Scholar

[28]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[29]

S. Zaremba, Sur un problème mixte relatif à l'équation de Laplace, (french) [On a mixed problem related to the Laplace equation], Bulletin international de l'Académie des Sciences de Cracovie. Classe des Sciences Mathématiques et Naturelles, Serie A: Sciences mathématiques (French), 313-344. Google Scholar

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