| 1. |
- Figalli, Alessio, et al.
(author)
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A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations
- 2014
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In: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 213:1, s. 269-286
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Journal article (peer-reviewed)abstract
- In this paper we study the fully nonlinear free boundary problem {F(D(2)u) = 1 almost everywhere in B-1 boolean AND Omega vertical bar D(2)u vertical bar <= K almost everywhhere in B-1\Omega, where K > 0, and Omega is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W (2,n) solutions are locally C (1,1) inside B (1). Under the extra condition that and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary partial derivative Omega boolean AND B-1.
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| 2. |
- Figalli, Alessio, et al.
(author)
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A general class of free boundary problems for fully nonlinear parabolic equations
- 2015
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In: Annali di Matematica Pura ed Applicata. - : Springer Science and Business Media LLC. - 0373-3114 .- 1618-1891. ; 194:4, s. 1123-1134
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Journal article (peer-reviewed)abstract
- In this paper, we consider the fully nonlinear parabolic free boundary problem { F(D(2)u) - partial derivative(t)u = 1 a.e. in Q(1) boolean AND Omega vertical bar D(2)u vertical bar + vertical bar partial derivative(t)u vertical bar <= K a.e. in Q(1)\Omega, where K > 0 is a positive constant, and Omega is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W-x(2,) (n) boolean AND W-t(1,) (n) solutions are locally C-x(1,) (1) boolean AND C-t(0,) (1) inside Q(1). A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result in Caffarelli and Huang (Duke Math J 118(1): 1-17, 2003). Once optimal regularity for u is obtained, we also show regularity for the free boundary partial derivative Omega boolean AND Q(1) under the extra condition that Omega superset of{u not equal 0}, and a uniform thickness assumption on the coincidence set {u = 0}.
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| 3. |
- Figalli, Alessio, et al.
(author)
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An overview of unconstrained free boundary problems
- 2015
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In: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 373:2050
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Journal article (peer-reviewed)abstract
- In this paper, we present a survey concerning unconstrained free boundary problems of type F-1(D(2)u, del u, u, x) = 0 in B-1 boolean AND Omega, F-2(D(2)u, del u, u, x) = 0 in B-1 \ Omega, u is an element of S(B-1), where B-1 is the unit ball, Omega is an unknown open set, F-1 and F-2 are elliptic operators (admitting regular solutions), and S is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.
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| 4. |
- Figalli, Alessio, et al.
(author)
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Constraint maps with free boundaries : the obstacle case
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Other publication (other academic/artistic)abstract
- This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As already pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, is the properties of the projected image and its higher regularity, which we show to be at most of class C2,1. In arbitrary dimensions, we prove that the image map is globally of class W3,BMO, and locally of class C2,1 around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.
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| 5. |
- Figalli, Alessio, et al.
(author)
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Constraint maps with free boundaries : the Bernoulli case
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Other publication (other academic/artistic)abstract
- In this manuscript, we delve into the study of maps that minimize the Alt–Caffarelli energy functional∫Ω(|Du|2 + q2χu−1(M )) dx,under the condition that the image u(Ω) is confined within ¯M . Here, Ω denotes a bounded domain in the ambient space Rn (with n ≥ 1), and M represents a smooth domain in the target space Rm (where m ≥ 2).Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, int(u−1(∂M )), such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary.Our first significant contribution is the validity of a ε-regularity theorem. This theorem is founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small energy.Our subsequent key finding reveals that, whenever the complement of M is uniformly convexand of class C3, the maps minimizing the Alt–Caffarelli energy with a positive parameter q exhibit Lipschitz continuity within a universally defined neighborhood of the non-coincidence set u−1(M ). In particular, this Lipschitz continuity extends to the free boundary.A noteworthy consequence of our findings is the smoothness of flat free boundaries and of theresulting image maps.
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| 6. |
- Figalli, Alessio, et al.
(author)
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Lipschitz regularity in vectorial linear transmission problems
- 2022
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In: Nonlinear Analysis. - : Elsevier BV. - 0362-546X .- 1873-5215. ; 221, s. 112911-
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Journal article (peer-reviewed)abstract
- We consider vector-valued solutions to a linear transmission problem, and we provethat Lipschitz-regularity on one phase is transmitted to the next phase. Moreexactly, given a solutionu:B1 subset of Rn -> Rmto the elliptic systemdiv((A+ (B-A)chi D) backward difference u) = 0inB1,whereAandBare Dini continuous, uniformly elliptic matrices, we prove that if backward difference u is an element of L infinity(D)thenuis Lipschitz inB1/2. A similar result is also derived for theparabolic counterpart of this problem.
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