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The issue of symmetry and symmetry breaking is fundamental in all areas ofscience. Symmetry is often assimilated to order and beauty while symmetrybreaking is the source of many interesting phenomena such as phase transitions,instabilities, segregation, self-organization, etc. In this contribution wereview a series of sharp results of symmetry of nonnegative solutions ofnonlinear elliptic differential equation associated with minimization problemson Euclidean spaces or manifolds. Nonnegative solutions of those equations areunique, a property that can also be interpreted as a rigidity result. Themethod relies on linear and nonlinear flows which reveal deep and robustproperties of a large class of variational problems. Local results on linearinstability leading to symmetry breaking and the bifurcation of non-symmetricbranches of solutions are reinterpreted in a larger, global, variationalpicture in which our flows characterize directions of descent.

Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs