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1. Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

   https://doi.org/10.1093/imrn/rnaa355  

   Bansil, Mohit; Kitagawa, Jun (December 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory. 

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2. 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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3. 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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4. The regular free boundary in the thin obstacle problem for degenerate parabolic equations

   Banerjee, A.; Danielli, D.; Garofalo, N.; Petrosyan, A. (June 2020, Algebra i analiz)

   In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial_t - \Delta_x)^s$$ for $$s \in (0,1)$$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $$H^{1+\gamma,\frac{1+\gamma}{2}}$$ regularity of the free boundary near such regular points. 

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5. Limit sets of unfolding paths in outer space

   https://doi.org/10.1017/S1474748023000488  

   Bestvina, Mladen; Gupta, Radhika; Tao, Jing (September 2024, Journal of the Institute of Mathematics of Jussieu)

   Abstract We construct an unfolding path in Outer space which does not converge in the boundary, and instead it accumulates on the entire 1-simplex of projectivized length measures on a nongeometric arational$${\mathbb R}$$-treeT. We also show thatTadmits exactly two dual ergodic projective currents. This is the first nongeometric example of an arational tree that is neither uniquely ergodic nor uniquely ergometric. 

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