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

Figallli, Alessio

doi:10.1142/9789813272880_0011

Cited by 8 publications

(6 citation statements)

References 19 publications

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“…This leads to a free boundary problem. Albeit classical (see [19,48] for an overview of the mathematical literature), the Stefan problem continues to be of use in many contemporary applications, such as additive manufacturing of alloys ( [32]), ice modeling for video rendering in computer graphics ( [31]), and, reaching even beyond its original fluid-mechanical nature, in the context of mathematical biology, for modeling the spread of various infectious diseases ( [14,36]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…(3) The obstacle problem. It is by now well-known that the classical Stefan problem (σ = 0), without source terms, is related to the parabolic obstacle problem through the so-called Duvaut transform (see [19,51] and the references therein). For control purposes, one could envisage transferring results from the Stefan problem to the parabolic obstacle problem (which is actually a problem to be studied in its own right, [20,51]).…”

Section: Epiloguementioning

confidence: 99%

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Control of the linearized Stefan problem in a periodic box

Geshkovski¹,

Maity²

2022

Preprint

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In this paper we consider the linearized one-phase Stefan problem with surface tension, set in the strip T × (−1, 1), thus with periodic boundary conditions respect to the horizontal direction x1 ∈ T. When the support of the control is not localized in x1, namely, is of the form ω = T × (c, d), we prove that the system is null-controllable in any positive time. We rely on a Fourier decomposition with respect to x1, and controllability results which are uniform with respect to the Fourier frequency parameter for the resulting family of one-dimensional systems. The latter results are also novel, as we compute the full spectrum of the underlying operator for the non-zero Fourier modes. The zeroth mode system, on the other hand, is seen as a controllability problem for the linear heat equation with a finite-dimensional constraint. We extend the controllability result to the setting of controls with a support localized in a box: ω = (a, b) × (c, d), through an argument inspired by the method of Lebeau and Robbiano, under the assumption that the initial data are of zero mean. Numerical experiments motivate several challenging open problems, foraying even beyond the specific setting we deal with herein.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Epiloguementioning

confidence: 99%

Control of the linearized Stefan problem in a periodic box

Geshkovski¹,

Maity²

2022

Preprint

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“…The main tool in deriving all such regularity properties are local monotonicity formulas and blow ups. We refer to [7], [11], [12], [22] and the references therein for more recent developments and extensions to more general operators.…”

Section: Context Of This Workmentioning

confidence: 99%

Qualitative properties of solutions to a non-local free boundary problem modeling cell polarization

Logioti¹,

Niethammer²,

Röger³

et al. 2022

Preprint

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We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction diffusion system of equations which models cell polarization. The authors have justified the well-posedness of this problem and have further proved uniqueness of solutions and global stability of steady states. In this paper we investigate qualitative properties of the free boundary. We present necessary and sufficient conditions for the initial data that imply continuity of the support at t = 0. If one of these assumptions fail, then jumps of the support take place. In addition we provide a complete characterization of the jumps for a large class of initial data.

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

Cited by 8 publications

References 19 publications

Control of the linearized Stefan problem in a periodic box

Control of the linearized Stefan problem in a periodic box

Qualitative properties of solutions to a non-local free boundary problem modeling cell polarization

A remark on memory effects in constrained fluid systems

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