UFPB's Webinar on Partial Differential Equations and Geometric Analysis
Event Period: Jul 28, 2020 a Dec 15, 2022
The purpose of this webinar is to maintain contact with researchers from around the world in the specific area of ​​nonlinear analysis, partial differential equations and geometric analysis. The talks will be held mainly on Tuesdays 4p.m. (GMT -3) via Google Meet.

Since 2020, we had a total of 42 talks by different researchers from all over the world. Please check our program for details and recordings of these talks. 

The 2022 season started on march 08. See the program for the confirmed schedule so far. 


João Henrique de Andrade (USP - Brazil)

17/05/2022 - 4p.m. local time (-3:00 GMT)

TitleMultiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds

AbstractWe prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation involving a multi-well (multiphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $\Gamma$-converges to the vectorial perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable set up to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to our geometric system are nondegenerate.

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