Symmetry breaking in problems involving semilinear equations

Previous Article
Discrete and differential homotopy in circular restricted three-body control
PROC Home
This Issue
Next Article
Non ordered lower and upper solutions to fourth order problems with functional boundary conditions

2011, 2011(Special): 219-228. doi: 10.3934/proc.2011.2011.219

Symmetry breaking in problems involving semilinear equations

Lucio Cadeddu ^1, and Giovanni Porru ^1,

1.	Dipartimento di Matematica e Informatica, Via Ospedale 72, 09124 Cagliari, Italy

Received July 2010 Revised March 2011 Published October 2011

Abstract
Related Papers
Under a Creative Commons license

This paper is concerned with two maximization problems where symmetry breaking arises. The rst one consists in the maximization of the energy integral relative to a homogeneous Dirichlet problem governed by the elliptic equation -$\deltau = XF^u^q$ in the annulus $B_(a,a+2)$ of the plane. Here 0 q < 1 and F is a varying subset of $B_(a,a+2)$, with a fixed measure. We prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough. The second problem we consider is governed by the same equation in a disc $B_a+2$ when $F$ varies in the annulus Ba;a+2 keeping a xed measure. So, now we have a so called maximization problem with a constraint. As in the previous case, we prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough.

Keywords: Symmetry breaking, Maximization, Rearrangements, Energy integral.

Mathematics Subject Classification: Primary: 49J20; Secondary: 35J25, 49K30.

Citation: Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219

[1]	Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161
[2]	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
[3]	Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61
[4]	Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112
[5]	Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[6]	Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
[7]	Linfeng Mei, Zongming Guo. Morse indices and symmetry breaking for the Gelfand equation in expanding annuli. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1509-1523. doi: 10.3934/dcdsb.2017072
[8]	Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129
[9]	Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[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]	Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic and Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002
[12]	Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341
[13]	Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[14]	Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure and Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415
[15]	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
[16]	Gabriela Kováčová, Birgit Rudloff, Igor Cialenco. Acceptability maximization. Frontiers of Mathematical Finance, 2022, 1 (2) : 219-248. doi: 10.3934/fmf.2021009
[17]	Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301
[18]	Naoki Shioji, Kohtaro Watanabe. Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4727-4770. doi: 10.3934/cpaa.2020210
[19]	Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054
[20]	Xavier Cabré, Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1179-1206. doi: 10.3934/dcds.2010.28.1179

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Lucio Cadeddu Giovanni Porru
on Google Scholar
Lucio Cadeddu Giovanni Porru

[Back to Top]