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1. Graphical solutions to one-phase free boundary problems

   https://doi.org/10.1515/crelle-2023-0067  

   Engelstein, Max; Fernández-Real, Xavier; Yu, Hui (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein’s problem for minimal surfaces.As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity. 

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2. Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

   https://doi.org/https://doi.org/10.1016/j.amc.2018.03.049  

   Hu, Bei-Bei; Xia, Tie-Cheng; Ma, Wen-Xiu (September 2018, Applied mathematics and computation)

   In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship. 

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3. The isoperimetric problem in the plane with the sum of two Gaussian densities

   https://doi.org/10.2140/involve.2018.11.549  

   Berry, J; Dannenberg, M; Liang, J; Zeng, Y (July 2018, Involve)

   We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We prove that the double Gaussian isoperimetric regions in the line are rays and that if the double Gaussian isoperimetric regions in the plane are half-spaces, then they must be bounded by vertical lines. 

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4. Fractional diffusion limit of a kinetic equation with diffusive boundary conditions in a bounded interval

   https://doi.org/10.3233/asy-221755  

   Cesbron, L.; Mellet, A.; Puel, M. (November 2022, Asymptotic Analysis)

   We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the unique solution of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established. 

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5. On the Line-Separable Unit-Disk Coverage and Related Problems

   https://doi.org/10.4230/LIPIcs.ISAAC.2023.51  

   Liu, Gang; Wang, Haitao (November 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Iwata, Satoru; Kakimura, Naonori (Ed.)

   Given a set P of n points and a set S of m disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line 𝓁. We present an m^{2/3} n^{2/3} 2^O(log^*(m+n)) + O((n+m)log(n+m)) time algorithm for the problem. This improves the previously best result of O(nm + n log n) time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of S are located on a line 𝓁 while points of P can be anywhere in the plane. Our algorithm runs in O(m√n + (n+m)log(n+m)) time, which improves the previously best result of O(nm log(m+n)) time. In addition, our results lead to an algorithm of n^{10/3} 2^O(log^*n) time for a half-plane coverage problem (given n half-planes and n points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of O(n⁴log n) time. Further, if all half-planes are lower ones, our algorithm runs in n^{4/3} 2^O(log^*n) time while the previously best algorithm takes O(n²log n) time. 

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