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1. Regularity of almost-minimizers of Hölder-coefficient surface energies

   https://doi.org/10.3934/dcds.2022015  

   Simmons, David A (January 2022, Discrete and Continuous Dynamical Systems)

   We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument. 

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2. Existence and regularity of minimizers for nonlocal energy functionals

   Foss, Mikil D.; Radu, Petronela; Wright, Cory (November 2019, Differential and integral equations)

   In this paper, we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics [19] or nonlocal diffusion models [1]. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure, we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory. 

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3. Branch points for (almost-)minimizers of two-phase free boundary problems

   https://doi.org/10.1017/fms.2022.105  

   David, Guy; Engelstein, Max; Smit Vega Garcia, Mariana; Toro, Tatiana (January 2023, Forum of Mathematics, Sigma)

   Abstract We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions. 

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4. Stability of periodic waves for the fractional KdV and NLS equations

   https://doi.org/10.1017/prm.2020.54  

   Hakkaev, Sevdzhan; Stefanov, Atanas G. (August 2021, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L 2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $$\lambda > 0$$ , there is a travelling wave solution to fKdV and fNLS $$\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $$ , which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in H α/2 [ − T , T ] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers. 

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5. Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers

   https://doi.org/10.1002/cpa.22184  

   Feldman, William M. (November 2023, Communications on Pure and Applied Mathematics)

   Abstract We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal homogenization theory in Lipschitz domains of Kenig et al. A key idea, to deal with the hard constraint on the volume, is a combination of a large scale almost dilation invariance with a selection principle argument. 

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