Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

Thomas, Marita

doi:10.3934/dcds.2015.35.2741

Cited by 7 publications

(3 citation statements)

References 36 publications

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“…The proof of Proposition 3.4 is carried out by contradiction to (3.21). The lower bound a(Γ C )ρ d−1 , which holds uniformly for all radii ρ , in every point of supp z k , is obtained with the aid of a uniform relative isoperimetric inequality deduced in [51,Thm. 3.2].…”

Section: Fine Properties Of the Solutionsmentioning

confidence: 99%

Stress-driven local-solution approach to quasistatic brittle delamination

Roubíček

Thomas

Panagiotopoulos

2015

Nonlinear Analysis: Real World Applications

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A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.

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Section: Fine Properties Of the Solutionsmentioning

confidence: 99%

Stress-driven local-solution approach to quasistatic brittle delamination

Roubíček

Thomas

Panagiotopoulos

2015

Nonlinear Analysis: Real World Applications

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“…Applying the co-area formula and the relative isoperimetric inequality (see for example [42]), we have ˆSi…”

Section: Estimate Of L N (I 0 ) and Existence Of The Free Boundarymentioning

confidence: 99%

Asymptotic behavior of the free interface for entire vector minimizers in phase transitions

Alikakos¹,

Geng²,

Zarnescu³

2021

Preprint

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We study globally bounded entire minimizers u : R n → R m of Allen-Cahn systems for potentials W ≥ 0 with {W = 0} = {a1, ..., aN } and W (u) ∼ |u − ai| α near u = ai, 0 < α < 2. Such solutions are, over large regions, identically equal to some zeroes of the potential ai's. We establish the estimatesfor the diffuse interface I0 := {x ∈ R n : min 1≤i≤N |u(x) − ai| > 0} and the free boundary ∂I0. Furthermore, if α = 1 we establish the upper bound

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“…Poincaré inequalities with uniform constants for classes of bounded and uniformly Lipschitz domains were discussed in [BC07], their Theorem 2 is a Lipschitz predecessor of our Theorem 3.1. Poincaré inequalities with uniform constants for domains satisfying certain cone conditions were proved in [Rui12] and [Tho15]. Optimal constants (in the sense of isoperimetry) for Poincaré inequalities involving trace terms on bounded Lipschitz domains were determined in [BGT19].…”

Section: Introductionmentioning

confidence: 99%

Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

Hinz¹,

Anna²,

Teplyaev³

2021

Preprint

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We study boundary value problems for bounded uniform domains in R n , n ≥ 2, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincaré inequalities with trace terms and uniform constants for uniform (ε, ∞)-domains within bounded common confinements. We then introduce generalized Dirichlet, Robin and Neumann problems for Poisson type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, and this also implies the norm convergence of the associated resolvents and the convergence of the corresponding eigenvalues and eigenfunctions. Based on our earlier work, we prove compactness results for parametrized classes of admissible domains, energy functionals and weak solutions. Using these results, we can verify the existence of optimal shapes in these classes.

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Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

Cited by 7 publications

References 36 publications

Stress-driven local-solution approach to quasistatic brittle delamination

Stress-driven local-solution approach to quasistatic brittle delamination

Asymptotic behavior of the free interface for entire vector minimizers in phase transitions

Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

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