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Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory
Blair Davey1, Mariana Smit Vega Garcia2
1Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717 USA.
Calculus of Variations and Partial Differential Equations
|January 30, 2024
Summary
This study extends a high-dimensional limiting technique to prove parabolic theorems for variable-coefficient heat operators. New proofs are provided for Carleman estimates and frequency function monotonicity, including a novel result.
Area of Science:
- Mathematical Analysis
- Partial Differential Equations
- Geometric Measure Theory
Background:
- A prior study introduced a high-dimensional limiting technique to connect parabolic and elliptic theorems.
- This technique was previously applied to constant-coefficient heat operators.
- Extending these methods to variable-coefficient operators presents new challenges.
Purpose of the Study:
- To generalize the high-dimensional limiting technique to the variable-coefficient setting for heat operators.
- To provide new proofs for established theorems concerning variable-coefficient heat equations.
- To establish a new monotonicity result for Alt-Caffarelli-Friedman-type functions in this context.
Main Methods:
- Application of a high-dimensional limiting technique, previously developed for constant-coefficient problems.
- Leveraging existing elliptic theorems as a foundation for proving parabolic counterparts.
- A limiting argument that transforms elliptic results into parabolic theorems for variable-coefficient operators.
Main Results:
- New proofs are presented for Carleman estimates for variable-coefficient heat operators.
- A new proof is provided for the monotonicity of Almgren-type frequency functions in the variable-coefficient setting.
- A novel monotonicity result for Alt-Caffarelli-Friedman-type functions is established and proven.
Conclusions:
- The high-dimensional limiting technique is effective for variable-coefficient parabolic problems.
- This approach simplifies proofs by relying on established elliptic theorems.
- The study contributes new theoretical results to the analysis of variable-coefficient heat operators.
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