Figure 1.
Examples of homogeneous solutions of the obstacle problem with obtuse/acute singular free boundary
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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
| 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 |
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.
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 and Continuous Dynamical Systems,
2018, 38
(12)
: 6073-6090.
doi: 10.3934/dcds.2018262
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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. |
show all references
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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