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1. Optimal Regularity and Structure of the Free Boundary for Minimizers in Cohesive Zone Models

   https://doi.org/10.1007/s00205-020-01509-3  

   Caffarelli, L.; Cagnetti, F.; Figalli, A. (July 2020, Archive for Rational Mechanics and Analysis)

   null (Ed.)

   Abstract We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $$C^{1, 1/2}$$ C 1 , 1 / 2 , and that near non-degenerate points the fracture set is $$C^{1, \alpha }$$ C 1 , α , for some $$\alpha \in (0, 1)$$ α ∈ ( 0 , 1 ) . 

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2. The Alt–Phillips functional for negative powers

   https://doi.org/10.1112/blms.12893  

   De_Silva, D; Savin, O (December 2023, Bulletin of the London Mathematical Society)

   Abstract We develop the free boundary regularity for nonnegative minimizers of the Alt–Phillips functional for negative power potentialsand establish a ‐convergence result of the rescaled energies to the perimeter functional as . 

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3. On the existence of non-flat profiles for a Bernoulli free boundary problem

   https://doi.org/10.1515/acv-2019-0063  

   Gravina, Giovanni; Leoni, Giovanni (January 2020, Advances in Calculus of Variations)

   null (Ed.)

   Abstract In this paper, we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions.We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt–Caffarelli energy functional. 

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4. Regularity of the free boundary for the two-phase Bernoulli problem

   https://doi.org/10.1007/s00222-021-01031-7  

   De Philippis, Guido; Spolaor, Luca; Velichkov, Bozhidar (August 2021, Inventiones mathematicae)

   null (Ed.)

   Abstract We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian. 

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5. Minimizers for the Cahn--Hilliard energy functional under strong anchoring conditions

   https://doi.org/10.1137/19M1309651  

   Dai, S.; Li, B.; Luong, T. (July 2020, SIAM journal on applied mathematics)

   null (Ed.)

   We study analytically and numerically the minimizers for the Cahn-Hilliard energy functional with a symmetric quartic double-well potential and under a strong anchoring condition(i.e., the Dirichlet condition) on the boundary of an underlying bounded domain. We show a bifurcation phenomenon determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. For the case that the boundary value is exactly the average of the two pure phases, if the bifurcation parameter is larger than or equal to a critical value, then the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. For a boundary value that is larger (or smaller) than that of the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain.Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian with the homogeneous boundary condition. We numerically minimize the functional E by solving the gradient-flow equation of E, i.e., the Allen-Cahn equation, with the designated boundary conditions, and with random initial values. We present our numerical simulations and discuss them in the context of our analytical results. 

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