Quantitative uniqueness of solutions to parabolic equations

Zhu, Jiuyi

doi:10.1016/j.jfa.2018.07.011

“…Since then, a wealth of unique continuation results for variable-coefficient heat operators have been established using both Carleman estimates and frequency functions. Recently, Carleman techniques have been used to establish space-like quantitative uniqueness of solutions to variable-coefficient parabolic equations that are averaged in time [56] and at a particular time-slice [7].…”

Section: Introductionmentioning

confidence: 99%

Parabolic Theory as a High-Dimensional Limit of Elliptic Theory

Davey

¹

2017

Arch Rational Mech Anal

1

11

0

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The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows. Specifically, we give new proofs of an L 2 Carleman estimate for the heat operator, and the monotonicity formulas for the frequency function associated to the heat operator, the two-phase free boundary problem, the flow of harmonic maps, and the mean curvature flow. The proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory. In particular, each parabolic theorem is proved by taking a highdimensional limit of the related elliptic result.

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“…Since then, a wealth of unique continuation results for variable-coefficient heat operators have been established using both Carleman estimates and frequency functions. Recently, Carleman techniques have been used to establish space-like quantitative uniqueness of solutions to variable-coefficient parabolic equations that are averaged in time [ 56 ] and at a particular time-slice [ 7 ].…”

Section: Introductionmentioning

confidence: 99%

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

Davey,

Smit Vega Garcia

2024

Calc. Var.

0

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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 → 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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“…In closing, we mention that there is a large literature on quantitative unique continuation. While we refer the reader to the introductions of [5] and [1] for a more detailed account, the following is a list of some of the most relevant works: [2], [3], [6], [7], [8], [17], [18], [21], [23], [24], [31], [32], [33], [44], [49], [51], [53] and [54].…”

Section: Introductionmentioning

confidence: 99%