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   https://doi.org/10.3233/ASY-211709  

   Niu, Weisheng; Shen, Zhongwei (May 2021, Asymptotic Analysis)

   null (Ed.)

   We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale κ that represents the strength of the singular perturbation and on the length scale ε of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale ε and independent of κ. This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both ε and κ. 

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2. Global well‐posedness for the one‐phase Muskat problem

   https://doi.org/10.1002/cpa.22124  

   Dong, Hongjie; Gancedo, Francisco; Nguyen, Huy Q. (December 2023, Communications on Pure and Applied Mathematics)

   Abstract The free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size. 

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3. Analysis of an hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

   https://doi.org/10.1051/m2an/2020015  

   Gong, Wei; Hu, Weiwei; Mateos, Mariano; Singler, John; Zhang, Yangwen (January 2020, ESAIM: Mathematical Modelling and Numerical Analysis)

   We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control.  The contribution of this paper is twofold.  First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain.  The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary.  Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution.  For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary.  We present numerical experiments to demonstrate the performance of the HDG method. 

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4. Optimal Domains for Elliptic Eigenvalue Problems with Rough Coefficients

   https://doi.org/10.1137/22M1523820  

   Snelson, Stanley; Teixeira, Eduardo V (June 2024, SIAM Journal on Mathematical Analysis)

   We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e., it is Lipschitz continuous at free boundary points. 

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5. 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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