Matteo Rizzi

The Allen-Cahn equation and general minimal cones

We construct solutions to the Allen-Cahn equation $\Delta u+u-u^3=0$ in $\mathbb{R}^{N+1}$, with $N\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying...