The Allen-Cahn equation and minimal surfaces

• Applicant: Dr. Matteo Rizzi, Ph.D.
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 521052394

The main purposes of the project are to construct new solutions to some semilinear elliptic PDEs, with particular interest in the Allen-Cahn equation, and to study the qualitative properties of a given solution. Our analysis will often rely on the theory of minimal (hyper)surfaces, which are crucial to understand the behaviour of the zero level set of solutions to the Allen-Cahn equation. A part of our project will be devoted to the fractional Allen-Cahn equation. We are particularly interested in k-ended solutions, that is entire solutions in the euclidean space whose zero level is given, outside a sufficiently large ball, by the disjoint union of a finite number of connected components. We have already constructed examples of such solutions in dimension N ≥ 8 enjoing some symmetry properties, namely they are O(m)×O(n)-invariant, m+n=N, with infinite Morse index and their energy on the ball has polynomial growth with respect to the radius. Our future goals are: -To investigate the qualitative properties of O(m)×O(n)-invariant solutions to the Allen-Cahn equation, with particular interest in their Morse index and in their zero level set. Their energy on the ball will play an important role. -To construct new O(m)×O(n)-invariant entire solutions to the Allen-Cahn equation in dimension 4 ≤ N ≤ 7 and new non O(m)×O(n)-invariant k-ended solutions in dimension N ≥ 8 (symmetry breaking solutions). Their zero level set will be prescribed, close to some suitable minimal hypersurface or to the union of k ≥ 2 graphs over such a hypersurface. -To prove an abstract result which could provide a general strategy to construct solutions to semilinear PDEs, related to nonvariational methods, such as the Lyapunov-Schmidt reduction. -To construct k-ended solutions to the fractional Allen-Cahn equation in dimension 3 whose zero level set is a union of normal graphs over some suitable minimal hypersurface.

• DFG Programme: Research Grants
• International Connection: Czech Republic
• Cooperation Partner: Dr. Oscar Ivan Agudelo Rico