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

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2024-01

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Springer Science and Business Media LLC

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 L2 → L2 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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elliptic theory, parabolic theory, high-dimensional limiting technique

Citation

Davey, B., Smit Vega Garcia, M. Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory. Calc. Var. 63, 40 (2024). https://doi.org/10.1007/s00526-023-02644-x

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