Same old, same old: Symmetries in the calculus of variations

Abstract:  Minimizing a functional requires its "first variation" to vanish for all compactly supported perturbations of the minimizer.  What if the "variation" in question arises from a symmetry of the functional, and isn't compactly supported?  Then even away from minimizers, the first variation depends only on the boundary of your spatial domain, so on minimizers, this boundary integral must vanish.  This is the content of Noether's Theorem for PDEs arising from a variational principle: there is a one-to-one correspondence between symmetries and vanishing boundary integrals (called conservation laws).  Because symmetries give you control over the set of solutions, conservation laws are connected with existence:  1. Minimizers of conserved integrals are invariant under the associated Noether symmetry group.  2. If conserved integrals arising from "large" symmetry groups make sense, then solutions don't exist.  3.  Sobolev norm minimizers are zero if the variational principle is dilation invariant (critical/supercritical exponents).