= Nonuniqueness of Dirichlet data without control at infinity
Unbounded-domain <Dirichlet problems> can need a condition at infinity. On all of $\mathbb R^n$, constant functions have identical empty boundary data and are distinct finite-energy stationary points of $\frac12\int|Du|^2$. With nonempty boundary, on the exterior of the unit ball in dimension at least three, $1-|x|^{2-n}$ is a nonzero harmonic function vanishing on the sphere, with finite <Dirichlet energy> and a nonzero limit at infinity. It is stationary for compactly supported variations. The permitted class of variations and the behaviour at infinity must be specified for a global variational uniqueness claim.
Back to article page
