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

Previous Article
Breathers as metastable states for the discrete NLS equation
DCDS Home
This Issue
Next Article
Arnold diffusion for a complete family of perturbations with two independent harmonics

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

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. 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

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Andrea Bonfiglioli, Ermanno Lanconelli and Francesco Uguzzoni. Levi's parametrix for some sub-elliptic non-divergence form operators. Electronic Research Announcements, 2003, 9: 10-18.
[2]	Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[3]	Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009
[4]	C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523
[5]	Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[6]	Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113
[7]	Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
[8]	Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[9]	Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271
[10]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[11]	Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[12]	Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[13]	Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1
[14]	Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[15]	Pablo Álvarez-Caudevilla, V. A. Galaktionov. Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators. Communications on Pure & Applied Analysis, 2016, 15 (1) : 261-286. doi: 10.3934/cpaa.2016.15.261
[16]	Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[17]	Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213
[18]	Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351
[19]	Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105
[20]	Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

2018 Impact Factor: 1.143

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.143

Tools

Figures and Tables

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors