Regularity and long-time behavior of nonlocal heat flows

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• 作者：Stanley Snelson
• 关键词：35K55 ; 49N60 ; 35B40 ; 35B25
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：54
• 期：2
• 页码：1705-1723
• 全文大小：557 KB
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• 作者单位：Stanley Snelson (1) (2)
  
  1. Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA
  2. Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL, 60637, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow approach to an optimal eigenvalue partition problem that, in light of the work of Caffarelli and Lin, is equivalent to a constrained harmonic mapping problem. In particular, we show that weak solutions of our heat equation converge as time approaches infinity to stationary solutions of this mapping problem. For these weak solutions, we also prove Lipschitz continuity and regularity of free interfaces, making use of a parabolic Almgren-type monotonicity formula. Mathematics Subject Classification 35K55 49N60 35B40 35B25