# American Institute of Mathematical Sciences

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

 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.

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:
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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. [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. [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. [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. [5] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184. doi: 10.1007/BF02392236. [6] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Providence: American Mathematical Society, 2005. doi: 10.1090/gsm/068. [7] F. Chiarenza, M. Frasca and P. Longo, W2, 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. [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. [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. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 1998. [11] A. Karakhanyan, Minimal Surfaces Arising in Singular Perturbation Problems, Preprint, 2016. [12] M. Kassmann, Harnack inequalities: An introduction, Bound. Value Probl., 2007 (2007), Art. ID 81415, 21 pp. [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. [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. [15] J. Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations, 8 (1983), 1605-1620. doi: 10.1080/03605308308820317. [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. [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. [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.
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