On the geometry and regularity of largest subsolutions for a free boundary problem in $$\mathbf{R }^2$$ : elliptic case

Orcan-Ekmekci, Betul

doi:10.1007/s00526-013-0606-8

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…Part (iv) is a straightforward barrier argument. See Orcan-Ekmekci [23] for nondegeneracy of largest subsolution in d = 2. Since the nondegeneracy of the maximal subsolution is not always a given, we will say that a maximal subsolution u is nondegenerate in a domain U if the estimate of Lemma 2.13 holds with a universal constant for every x ∈ ∂{u > 0} ∩ U and ball B r (x) ⊂ U .…”

Section: 3mentioning

confidence: 99%

Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media

Feldman¹

2018

Preprint

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This paper considers the Alt-Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction e ∈ S d−1 . In d = 2 we characterize the regularity properties of the pinning interval in terms of the normal direction, including possible discontinuities at rational directions. These results require a careful study of the families of plane-like solutions available with a given slope. Using the same techniques we also obtain strong, in some cases optimal, bounds on the class of limit shapes of local minimizers in d = 2, and preliminary results in d ≥ 3.

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Section: 3mentioning

confidence: 99%

Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media

Feldman¹

2018

Preprint

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“…the free boundary being the frontier of the positivity set {u > 0} (see Figure 1). A huge literature about free boundaries stemmed from the existence and regularity results proved in [2] (without any attempt of completeness, see for instance [3,22,20,21,30,33]). However, these papers are mainly focused on the study of local minimizers, through the Euler-Lagrange equation and the related free boundary condition, intended in the variational or in the viscosity sense.…”

Section: Introductionmentioning

confidence: 99%

A Duality Theory for Non-convex Problems in the Calculus of Variations

Bouchitté

Fragalà

2018

Arch Rational Mech Anal

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We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min-max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in [11].

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Limit Shapes of Local Minimizers for the Alt–Caffarelli Energy Functional in Inhomogeneous Media

Feldman

2021

Arch Rational Mech Anal

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On the geometry and regularity of largest subsolutions for a free boundary problem in $$\mathbf{R }^2$$ : elliptic case

Cited by 3 publications

References 11 publications

Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media

Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media

A Duality Theory for Non-convex Problems in the Calculus of Variations

Limit Shapes of Local Minimizers for the Alt–Caffarelli Energy Functional in Inhomogeneous Media

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