Symmetry and asymptotics in semilinear elliptic problems

Consider a domain of Euclidean space having a certain symmetry, say about a point or a hyperplane. Is it true that every solution of a
nonlinear PDE inherits the symmetry property, if the equation allows it? Starting from a classical result of Gidas, Ni and Nirenberg, we
question the optimality of the conditions under which symmetry is preserved/violated, in the following directions: weak solutions, nodal
solutions, non-lipschitz nonlinearities.
I will try to make a case for the asymptotic nature of the problem, by spending the second part of the talk on the case of boundary blow-up
solutions. The proofs rely on classical tools, such as the maximum principle and
the Alexandrov device of moving parallel hyperplanes, as well as a more recent technique developped with O. Costin (Ohio State U),
stemming from exponential asymptotics.

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