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- Caffarelli, Luis A., et al.
(författare)
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A MINIMIZATION PROBLEM WITH FREE BOUNDARY RELATED TO A COOPERATIVE SYSTEM
- 2018
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Ingår i: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 167:10, s. 1825-1882
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Tidskriftsartikel (refereegranskat)abstract
- We study the minimum problem for the functional integral(Omega)(vertical bar del u vertical bar(2) + Q(2) chi({vertical bar u vertical bar>0}))dx with the constraint u(i) >= 0 for i = 1,... , m, where Omega subset of R-n is a bounded domain and u = (u(1),... , u(m)) is an element of H-1 (Omega;R-m). First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set {vertical bar u vertical bar > 0} is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary Omega boolean AND partial derivative{vertical bar u vertical bar> 0}.
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| 2. |
- Caffarelli, Luis A., et al.
(författare)
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Regularity of free boundaries a heuristic retro
- 2015
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Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 373:2050
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Forskningsöversikt (refereegranskat)abstract
- This survey concerns regularity theory of a few free boundary problems that have been developed in the past half a century. Our intention is to bring up different ideas and techniques that constitute the fundamentals of the theory. We shall discuss four different problems, where approaches are somewhat different in each case. Nevertheless, these problems can be divided into two groups: (i) obstacle and thin obstacle problem; (ii) minimal surfaces, and cavitation flow of a perfect fluid. In each case, we shall only discuss the methodology and approaches, giving basic ideas and tools that have been specifically designed and tailored for that particular problem. The survey is kept at a heuristic level with mainly geometric interpretation of the techniques and situations in hand.
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