Optimal Besov Differentiability for Entropy Solutions of the Eikonal Equation

Ghiraldin, Francesco; Lamy, Xavier

doi:10.1002/cpa.21868

Cited by 25 publications

(42 citation statements)

References 43 publications

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“…A fundamental tool in the study of fine properties of elements of A( ) is the kinetic formulation [18] (see also [23] in the framework of scalar conservation laws). Here we use a more recent version obtained in [13].…”

Section: Previous Resultsmentioning

confidence: 99%

“…We observe that if σ solves (1.7), then σ + μ ⊗ L 1 also solves (1.7) for every μ ∈ M loc ( ). This ambiguity is resolved in [13] by considering the unique σ 0 solving (1.7) such that…”

Section: Previous Resultsmentioning

confidence: 99%

“…Notice that ∈ E π if and only if ψ is π -periodic. Rephrasing the construction in [13], we have the following identity: for every ∈ E π and every…”

Section: Previous Resultsmentioning

confidence: 99%

“…It was shown in [13] that the constraint forces σ to be π -periodic in s, in particular for L 1 × ν min -a.e. (t, x) ∈ (0, T ) × the support of ∂ s σ ω is π -periodic.…”

Section: By Construction We Havementioning

confidence: 99%

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Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

Marconi

2021

Arch Rational Mech Anal

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We consider the singularly perturbed problem $$F_\varepsilon (u,\Omega ):=\int _\Omega \varepsilon |\nabla ^2u|^2 + \varepsilon ^{-1}|1-|\nabla u|^2|^2$$ F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if $$\Omega $$ Ω is an ellipse, then the minimizers of $$F_\varepsilon (\cdot ,\Omega )$$ F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation $$|\nabla u|=1$$ | ∇ u | = 1 as $$\varepsilon \rightarrow 0$$ ε → 0 .

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Section: Previous Resultsmentioning

confidence: 99%

Section: Previous Resultsmentioning

confidence: 99%

“…Notice that ∈ E π if and only if ψ is π -periodic. Rephrasing the construction in [13], we have the following identity: for every ∈ E π and every…”

Section: Previous Resultsmentioning

confidence: 99%

“…It was shown in [13] that the constraint forces σ to be π -periodic in s, in particular for L 1 × ν min -a.e. (t, x) ∈ (0, T ) × the support of ∂ s σ ω is π -periodic.…”

Section: By Construction We Havementioning

confidence: 99%

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Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

Marconi

2021

Arch Rational Mech Anal

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“…In particular, it is possible to define a countably-rectifiable jump set J u with weak traces u ± on both sides. Probably the main open question to complete the proof of the Γ−convergence is to show that the entropy production is concentrated on J u , see [GL20,Mar20,Mar21,LP21] for recent progress on this question.…”

Section: Introductionmentioning

confidence: 99%

Compactness and structure of zero-states for unoriented Aviles-Giga functionals

Goldman,

Merlet,

Pegon

et al. 2021

Preprint

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Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional.We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.

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