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We construct an entire solution U : R 2 → R 2 $U:\mathbb {R}^2\rightarrow \mathbb {R}^2$ to the elliptic system Δ U = ∇ u W ( U ) , $$\begin{equation*} \Delta U=\nabla _uW(U), \end{equation*}$$ where W : R 2 → [ 0 , ∞ ) $W:\mathbb {R}^2\rightarrow [0,\infty)$ is a ‘triple-well’ potential.
We construct an entire solution $U:\mathbb{R}^2\to\mathbb{R}^2$ to the elliptic system \[ ΔU=\nabla_uW(U), \] where $W:\mathbb{R}^2\to [0,\infty)$ is a `triple-well' potential. This solution is a local minimizer of the associated energy \[ \int \frac{1}{2}|\nabla U|^2+W(U)\,dx \] in the sense that $U$ minimizes the energy on any compact set among competitors agreeing with $U$ outside that set.
Peter Sternberg speaking at BIRS workshop, Phase Transitions Models, on Tuesday, May 2, 2017 on the topic: A Degenerate Isoperimetric Problem in the Plane.
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