Free boundary regularity in obstacle problems

Figalli, Alessio

doi:10.5802/jedp.662

Cited by 7 publications

(9 citation statements)

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“…We refer to [9,10] for further theoretical results on difficult and actual issues raised by the obstacle problem such as: the regularity of the interface, analysis of the blow up at singular points and extensions to the timeevolution case (parabolic obstacle problem). The rest of this subsection is devoted to formal rewritings of the obstacle problem.…”

Section: Canonical Example: the Obstacle Problemmentioning

confidence: 99%

A remark on memory effects in constrained fluid systems

Perrin¹

2020

ESAIM: ProcS

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The goal of this note is to put into perspective the recent results obtained on memory effects in partially congested fluid systems of Euler or Navier-Stokes type with former studies on free boundary obstacle problems and Hele-Shaw equations. In particular, we relate the notion of adhesion potential initially introduced in the context of dense suspension flows with the one of Baiocchi variable used in the analysis of free boundary problems.

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Section: Canonical Example: the Obstacle Problemmentioning

confidence: 99%

A remark on memory effects in constrained fluid systems

Perrin¹

2020

ESAIM: ProcS

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“…see for instance [16,Section 3]. In particular, up to replacing u by u/g we can assume that g = 1, which shows that (2.1) corresponds to the Euler-Lagrange equations associated to the minimization problem (2.3).…”

Section: The Elliptic Obstacle Problemmentioning

confidence: 99%

“…A key result in the theory of obstacle problems states that the estimate above holds even for p = ∞, see [19,5,7,16]:…”

Section: The Elliptic Obstacle Problemmentioning

confidence: 99%

“…If we represents the membrane as the graph of a nonnegative function u : Ω → R, this function minimizes the functional (here g > 0 is the gravitational constant), see Figure 6. The existence and uniqueness of a minimizer for (2.3) follows by standard techniques in the calculus of variations, see for instance [16,Section 2]. Then, computing the Euler-Lagrange equations for the minimizer u, one can prove that u satisfies the equation ∆u = g χ {u>0} , see for instance [16,Section 3].…”

Section: The Elliptic Obstacle Problemmentioning

confidence: 99%

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Regularity of Interfaces in Phase Transitions via Obstacle Problems - Fields Medal Lecture

Figallli

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems.After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Schaeffer's conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.

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“…The key idea is to exploit the degeneracy of the differential operator at the boundary to construct (logarithmically) diverging barriers. On a different note, regarding the regularity of the solution u across the free boundary and the more subtle question of the regularity of the free boundary itself, we refer the interested reader to the crucial work of Caffarelli [1], and the expository notes [3].…”

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confidence: 99%