SYMMETRY AND SYMMETRY BREAKING: RIGIDITY AND FLOWS IN ELLIPTIC PDES
SYMMETRY AND SYMMETRY BREAKING: RIGIDITY AND FLOWS IN ELLIPTIC PDES
复制标题
对称性和对称性破缺：椭圆 PDES 中的刚性和流动
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
M. Esteban
中科院分区：
文献类型：
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作者：
J. Dolbeault;M. Esteban;M. Loss;M. Esteban
对称性和对称性破缺问题是所有科学领域的基础。对称性常常被视为秩序和美，而对称性破缺则是许多有趣现象的根源，例如相变、不稳定性、偏析、自组织等。在本文中，我们回顾了与欧几里德空间或流形上的最小化问题相关的非线性椭圆微分方程非负解的对称性的一系列尖锐结果。这些方程的非负解是唯一的，这一属性也可以解释为刚性结果。该方法依赖于线性和非线性流，揭示了一大类变分问题的深层和鲁棒特性。导致对称性破缺和解的非对称分支分叉的线性不稳定性的局部结果在更大的全局变分图中被重新解释，其中我们的流表征了下降方向。
The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions, instabilities, segregation, self-organization, etc. In this contribution we review a series of sharp results of symmetry of nonnegative solutions of nonlinear elliptic differential equation associated with minimization problems on Euclidean spaces or manifolds. Nonnegative solutions of those equations are unique, a property that can also be interpreted as a rigidity result. The method relies on linear and nonlinear flows which reveal deep and robust properties of a large class of variational problems. Local results on linear instability leading to symmetry breaking and the bifurcation of non-symmetric branches of solutions are reinterpreted in a larger, global, variational picture in which our flows characterize directions of descent.