Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

Serena Dipierro; Aram Karakhanyan; Enrico Valdinoci

doi:10.3934/dcds.2018262

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December 2018, 38(12): 6073-6090. doi: 10.3934/dcds.2018262

Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

Serena Dipierro ^1, , Aram Karakhanyan ^2, and Enrico Valdinoci ^{1, 3, 4,,}

1.	School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia
2.	Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
3.	Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy
4.	Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133 Milan, Italy

* Corresponding author: Enrico Valdinoci

Received July 2017 Revised February 2018 Published September 2018

Fund Project: Supported by INdAM Istituto Nazionale di Alta Matematica and Australian Research Council Discovery Project DP170104880 NEW Nonlocal Equations at Work

We provide an integral estimate for a non-divergence (non-varia-tional) form second order elliptic equation $a_{ij}u_{ij} = u^p$, $u≥ 0$, $p∈[0, 1)$, with bounded discontinuous coefficients $a_{ij}$ having small BMO norm. We consider the simplest discontinuity of the form $x\otimes x|x|^{-2}$ at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when $p = 0$) cannot be smooth at the points of discontinuity of $a_{ij}(x)$.

To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

Keywords: Free boundary, blow-up sequences, non-divergence operators, monotonicity formulae.

Mathematics Subject Classification: 35R35, 35B65.

Citation: Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262

References:

[1]	H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107. Google Scholar
[2]	D. J. Araújo, R. Leitão and E. V. Teixeira, Infinity Laplacian equation with strong absorptions, J. Funct. Anal., 270 (2016), 2249-2267. doi: 10.1016/j.jfa.2015.12.012. Google Scholar
[3]	I. Blank and K. Teka, The Caffarelli alternative in measure for the nondivergence form elliptic obstacle problem with principal coefficients in VMO, Comm. Partial Differential Equations, 39 (2014), 321-353. doi: 10.1080/03605302.2013.823988. Google Scholar
[4]	X. Cabré, Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions, Discrete Contin. Dyn. Syst., 20 (2008), 425-457. doi: 10.3934/dcds.2008.20.425. Google Scholar
[5]	L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184. doi: 10.1007/BF02392236. Google Scholar
[6]	L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Providence: American Mathematical Society, 2005. doi: 10.1090/gsm/068. Google Scholar
[7]	F. Chiarenza, M. Frasca and P. Longo, W^{2, p}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. doi: 10.2307/2154379. Google Scholar
[8]	S. Dipierro, O. Savin and E. Valdinoci, A nonlocal free boundary problem, SIAM J. Math. Anal., 47 (2015), 4559-4605. doi: 10.1137/140999712. Google Scholar
[9]	M. Focardi, M. S. Gelli and E. Spadaro, Monotonicity formulas for obstacle problems with Lipschitz coefficients, Calc. Var. Partial Differential Equations, 54 (2015), 1547-1573. doi: 10.1007/s00526-015-0835-0. Google Scholar
[10]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 1998. Google Scholar
[11]	A. Karakhanyan, Minimal Surfaces Arising in Singular Perturbation Problems, Preprint, 2016. Google Scholar
[12]	M. Kassmann, Harnack inequalities: An introduction, Bound. Value Probl., 2007 (2007), Art. ID 81415, 21 pp. Google Scholar
[13]	A. Petrosyan, H. Shahgholian and N. Uraltseva, Regularity of Free Boundaries in Obstacle-Type Problems, Providence: American Mathematical Society, 2012. doi: 10.1090/gsm/136. Google Scholar
[14]	D. dos Prazeres and E. V. Teixeira, Cavity problems in discontinuous media, Calc. Var. Partial Differential Equations, 55 (2016), Art. 10, 15 pp. doi: 10.1007/s00526-016-0955-1. Google Scholar
[15]	J. Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations, 8 (1983), 1605-1620. doi: 10.1080/03605308308820317. Google Scholar
[16]	E. V. Teixeira, Hessian continuity at degenerate points in nonvariational elliptic problems, Int. Math. Res. Not. IMRN, (2015), 6893-6906. doi: 10.1093/imrn/rnu150. Google Scholar
[17]	E. V. Teixeira, Regularity for the fully nonlinear dead-core problem, Math. Ann., 364 (2016), 1121-1134. doi: 10.1007/s00208-015-1247-3. Google Scholar
[18]	N. Trudinger, Elliptic equations in non-divergence form, Miniconference on Partial Differential Equations(Canberra, 1981), Proc. Centre Math. Anal. Austral. Nat. Univ., 1, Austral. Nat. Univ., Canberra, 1982, 1-16. Google Scholar

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References:

[1]	H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107. Google Scholar
[2]	D. J. Araújo, R. Leitão and E. V. Teixeira, Infinity Laplacian equation with strong absorptions, J. Funct. Anal., 270 (2016), 2249-2267. doi: 10.1016/j.jfa.2015.12.012. Google Scholar
[3]	I. Blank and K. Teka, The Caffarelli alternative in measure for the nondivergence form elliptic obstacle problem with principal coefficients in VMO, Comm. Partial Differential Equations, 39 (2014), 321-353. doi: 10.1080/03605302.2013.823988. Google Scholar
[4]	X. Cabré, Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions, Discrete Contin. Dyn. Syst., 20 (2008), 425-457. doi: 10.3934/dcds.2008.20.425. Google Scholar
[5]	L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184. doi: 10.1007/BF02392236. Google Scholar
[6]	L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Providence: American Mathematical Society, 2005. doi: 10.1090/gsm/068. Google Scholar
[7]	F. Chiarenza, M. Frasca and P. Longo, W^{2, p}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. doi: 10.2307/2154379. Google Scholar
[8]	S. Dipierro, O. Savin and E. Valdinoci, A nonlocal free boundary problem, SIAM J. Math. Anal., 47 (2015), 4559-4605. doi: 10.1137/140999712. Google Scholar
[9]	M. Focardi, M. S. Gelli and E. Spadaro, Monotonicity formulas for obstacle problems with Lipschitz coefficients, Calc. Var. Partial Differential Equations, 54 (2015), 1547-1573. doi: 10.1007/s00526-015-0835-0. Google Scholar
[10]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 1998. Google Scholar
[11]	A. Karakhanyan, Minimal Surfaces Arising in Singular Perturbation Problems, Preprint, 2016. Google Scholar
[12]	M. Kassmann, Harnack inequalities: An introduction, Bound. Value Probl., 2007 (2007), Art. ID 81415, 21 pp. Google Scholar
[13]	A. Petrosyan, H. Shahgholian and N. Uraltseva, Regularity of Free Boundaries in Obstacle-Type Problems, Providence: American Mathematical Society, 2012. doi: 10.1090/gsm/136. Google Scholar
[14]	D. dos Prazeres and E. V. Teixeira, Cavity problems in discontinuous media, Calc. Var. Partial Differential Equations, 55 (2016), Art. 10, 15 pp. doi: 10.1007/s00526-016-0955-1. Google Scholar
[15]	J. Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations, 8 (1983), 1605-1620. doi: 10.1080/03605308308820317. Google Scholar
[16]	E. V. Teixeira, Hessian continuity at degenerate points in nonvariational elliptic problems, Int. Math. Res. Not. IMRN, (2015), 6893-6906. doi: 10.1093/imrn/rnu150. Google Scholar
[17]	E. V. Teixeira, Regularity for the fully nonlinear dead-core problem, Math. Ann., 364 (2016), 1121-1134. doi: 10.1007/s00208-015-1247-3. Google Scholar
[18]	N. Trudinger, Elliptic equations in non-divergence form, Miniconference on Partial Differential Equations(Canberra, 1981), Proc. Centre Math. Anal. Austral. Nat. Univ., 1, Austral. Nat. Univ., Canberra, 1982, 1-16. Google Scholar

Figure 1. Examples of homogeneous solutions of the obstacle problem with obtuse/acute singular free boundary

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Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

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