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1. Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory

   https://doi.org/10.1007/s00526-023-02644-x  

   Davey, Blair; Smit Vega Garcia, Mariana (January 2024, Calculus of Variations and Partial Differential Equations)

   Abstract This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159–196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of$$L^2 \rightarrow L^2$$ $L^2\to L^2$Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt–Caffarelli–Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result. 

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2. Relationships between Global and Local Monotonicity of Operator

   Khanh, PD; Khoa, VVH; Martinez-Legaz, JE; Mordukhovich, BS (January 2025, Journal of Convex Analysis)

   The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between fnite-dimensional and infnite-dimensional spaces. We frst show that for single-valued operators with convex domains in locally convex topological spaces, their continuity ensures that their global monotonicity agrees with the local one around any point of the graph. This also holds for set-valued mappings defned on the real line under a certain connectedness condition. The situation is diferent for set-valued operators in multidimensional spaces as demonstrated by an example of locally monotone operator on the plane that is not globally monotone. Finally, we invoke coderivative criteria from variational analysis to characterize both global and local maximal monotonicity of set-valued operators in Hilbert spaces to verify the equivalence between these monotonicity properties under the closedgraph and global hypomonotonicity assumptions. 

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3. Elliptic zastava

   https://doi.org/10.1090/jag/803  

   Finkelberg, Michael; Matviichuk, Mykola; Polishchuk, Alexander (April 2023, Journal of Algebraic Geometry)

   We study the elliptic zastava spaces, their versions (twisted, Coulomb, Mirković local spaces, reduced) and relations with monowalls moduli spaces and Feigin-Odesskiĭ moduli spaces of G G -bundles with parabolic structure on an elliptic curve. 

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4. The geometry of monotone operator splitting methods

   https://doi.org/10.1017/S0962492923000065  

   Combettes, Patrick L (July 2024, Acta Numerica)

   We propose a geometric framework to describe and analyse a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones. 

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5. Parabolic Frequency on Manifolds

   https://doi.org/10.1093/imrn/rnab052  

   Holck Colding, Tobias; Minicozzi II, William P (April 2021, International Mathematics Research Notices)

   Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness. 

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