A computer assisted proof of the symmetry of solutions to a PDE

The Curie symmetry principle, which asserts that the symmetries of the cause must also be contained in the effects, no longer holds in general for non-linear partial differential equations (PDE) with possibly infinitely many solutions. In this context, one has to restrict to 'low energy' solutions. Solutions with the lowest energy (ground states) usually - though not always - possess all the symmetries of the problem. For solutions with the lowest energy among sign-changing ones, only part of the symmetries is preserved. In this talk, I will consider the simple problem -Δu = |u|ᵖ⁻²u with zero Dirichlet boundary conditions on a square domain and show how numerical computations help to discover the residual symmetry of the least-energy sign-changing solutions as well as to prove that what the simulation tells is rigorously valid.