-
Previous Article
Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms
- CPAA Home
- This Issue
-
Next Article
Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities
A symmetry result for elliptic systems in punctured domains
1. | Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica della Marche, Via Brecce Bianche, 60131, Ancona, Italy |
2. | Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009 Crawley, Australia |
3. | Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38123, Povo (Trento), Italy |
We consider an elliptic system of equations in a punctured bounded domain. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperplane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction. As a consequence, we prove symmetry results also for a related polyharmonic problem of any order with Navier boundary conditions.
References:
[1] |
D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer-Verlag London, Ltd., London, 2001.
doi: 10.1007/978-1-4471-0233-5. |
[2] |
E. Berchio, F. Gazzola and T. Weth,
Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., 620 (2008), 165-183.
doi: 10.1515/CRELLE.2008.052. |
[3] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N. S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[4] |
L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. Ⅱ. Symmetry and monotonicity via moving planes, Advances in Geometric Analysis, Int. Press, Somerville, MA, 21 (2012), 97–105. |
[5] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
|
[6] |
F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math., 21 (2019) no.2, 1850019, 34 pp.
doi: 10.1142/S0219199718500190. |
[7] |
F. Colasuonno and E. Vecchi,
Symmetry and rigidity in the hinged composite plate problem, J. Differential Equations, 266 (2019), 4901-4924.
doi: 10.1016/j.jde.2018.10.011. |
[8] |
L. Damascelli and F. Pacella,
Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal., 45 (2013), 1003-1026.
doi: 10.1137/110853534. |
[9] |
D. G. De Figueiredo,
Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123.
doi: 10.1007/BF01193947. |
[10] |
F. Esposito, A. Farina and B. Sciunzi,
Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983.
doi: 10.1016/j.jde.2018.04.030. |
[11] |
A. Ferrero, F. Gazzola and T. Weth,
Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl., 186 (2007), 565-578.
doi: 10.1007/s10231-006-0019-9. |
[12] |
F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010, 1991.
doi: 10.1007/978-3-642-12245-3. |
[13] |
B. Gidas, B, W. M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
|
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. |
[15] |
L. Montoro, F. Punzo and B. Sciunzi,
Qualitative properties of singular solutions to nonlocal problems, Ann. Mat. Pura Appl., 197 (2018), 941-964.
doi: 10.1007/s10231-017-0710-z. |
[16] |
P. Pizzetti,
Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rendiconti Lincei, 18 (1909), 182-185.
|
[17] |
B. Sciunzi,
On the moving plane method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl., 108 (2017), 111-123.
doi: 10.1016/j.matpur.2016.10.012. |
[18] |
J. Serrin,
A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
doi: 10.1007/BF00250468. |
[19] |
S. Terracini,
On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.
|
[20] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
show all references
References:
[1] |
D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer-Verlag London, Ltd., London, 2001.
doi: 10.1007/978-1-4471-0233-5. |
[2] |
E. Berchio, F. Gazzola and T. Weth,
Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., 620 (2008), 165-183.
doi: 10.1515/CRELLE.2008.052. |
[3] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N. S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[4] |
L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. Ⅱ. Symmetry and monotonicity via moving planes, Advances in Geometric Analysis, Int. Press, Somerville, MA, 21 (2012), 97–105. |
[5] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
|
[6] |
F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math., 21 (2019) no.2, 1850019, 34 pp.
doi: 10.1142/S0219199718500190. |
[7] |
F. Colasuonno and E. Vecchi,
Symmetry and rigidity in the hinged composite plate problem, J. Differential Equations, 266 (2019), 4901-4924.
doi: 10.1016/j.jde.2018.10.011. |
[8] |
L. Damascelli and F. Pacella,
Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal., 45 (2013), 1003-1026.
doi: 10.1137/110853534. |
[9] |
D. G. De Figueiredo,
Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123.
doi: 10.1007/BF01193947. |
[10] |
F. Esposito, A. Farina and B. Sciunzi,
Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983.
doi: 10.1016/j.jde.2018.04.030. |
[11] |
A. Ferrero, F. Gazzola and T. Weth,
Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl., 186 (2007), 565-578.
doi: 10.1007/s10231-006-0019-9. |
[12] |
F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010, 1991.
doi: 10.1007/978-3-642-12245-3. |
[13] |
B. Gidas, B, W. M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
|
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. |
[15] |
L. Montoro, F. Punzo and B. Sciunzi,
Qualitative properties of singular solutions to nonlocal problems, Ann. Mat. Pura Appl., 197 (2018), 941-964.
doi: 10.1007/s10231-017-0710-z. |
[16] |
P. Pizzetti,
Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rendiconti Lincei, 18 (1909), 182-185.
|
[17] |
B. Sciunzi,
On the moving plane method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl., 108 (2017), 111-123.
doi: 10.1016/j.matpur.2016.10.012. |
[18] |
J. Serrin,
A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
doi: 10.1007/BF00250468. |
[19] |
S. Terracini,
On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.
|
[20] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[1] |
Giuseppe Riey, Berardino Sciunzi. One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1157-1166. doi: 10.3934/cpaa.2012.11.1157 |
[2] |
Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045 |
[3] |
Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022 |
[4] |
Phuong Le, Hoang-Hung Vo. Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1027-1048. doi: 10.3934/cpaa.2022008 |
[5] |
Serena Dipierro. Geometric inequalities and symmetry results for elliptic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473 |
[6] |
Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886 |
[7] |
Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 |
[8] |
Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051 |
[9] |
Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079 |
[10] |
Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure and Applied Analysis, 2022, 21 (3) : 837-844. doi: 10.3934/cpaa.2021201 |
[11] |
Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425 |
[12] |
Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 |
[13] |
Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41 |
[14] |
Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505 |
[15] |
Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151 |
[16] |
Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 |
[17] |
Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 |
[18] |
Xueying Chen, Guanfeng Li, Sijia Bao. Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1755-1772. doi: 10.3934/cpaa.2022045 |
[19] |
Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 141-159. doi: 10.3934/dcdsb.2019176 |
[20] |
Giovany Figueiredo, Marcelo Montenegro, Matheus F. Stapenhorst. A log–exp elliptic equation in the plane. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 481-504. doi: 10.3934/dcds.2021125 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]