# Applied Mathematics Colloquium: Dr. Nick Alikakos

## University of Athens

Friday, April 29, 2022

2:00 PM – 3:00 PM

2:00 PM – 3:00 PM

Title: Sharp Lower Bounds for the Vector Allen-Cahn Energy and Qualitative

Properties of Minimizers

Abstract: We study minimizers Uε of J(U)=Integral over Ω of (ε|grad U| + (1/ε)W(U))dx, W>0 on Rm\{a1,..., aN}, for bounded domains Ω in R2, with certain geometrical features, and prepared Dirichlet data. We derive a sharp lower bound, as ε tends to 0, with the additional feature that it involves half of the gradient and part of the domain. Based on this we derive very precise (in ε) pointwise estimates up to the boundary of the domain, and also up to the boundary of the limiting partition. We do not impose symmetry hypotheses and we do not employ Γ-convergence techniques. Our results, via rescaling, have implications on the existence of entire solutions connecting global minima of W.

Properties of Minimizers

Abstract: We study minimizers Uε of J(U)=Integral over Ω of (ε|grad U| + (1/ε)W(U))dx, W>0 on Rm\{a1,..., aN}, for bounded domains Ω in R2, with certain geometrical features, and prepared Dirichlet data. We derive a sharp lower bound, as ε tends to 0, with the additional feature that it involves half of the gradient and part of the domain. Based on this we derive very precise (in ε) pointwise estimates up to the boundary of the domain, and also up to the boundary of the limiting partition. We do not impose symmetry hypotheses and we do not employ Γ-convergence techniques. Our results, via rescaling, have implications on the existence of entire solutions connecting global minima of W.

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