Tangential Touch Between Free and Fixed Boundaries in a Problem from Superconductivity

Matevosyan, Norayr

doi:10.1080/03605300500257628

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…The evolution of the level set function ϕ as well as the decrease of the energy functional is illustrated in Figure 1. Note that the moving boundary touches the fixed one tangentially, which is a good check with theoretical prediction (see [21]). One also observes the expected decrease and final (approximate) stationarity of the energy functional.…”

Section: Numerical Solutionsupporting

confidence: 83%

A Level Set Based Shape Optimization Method for an Elliptic Obstacle Problem

Burger

Matevosyan

Wolfram

2011

Math. Models Methods Appl. Sci.

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In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape.Using the shape sensitivities we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem.Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.

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Section: Numerical Solutionsupporting

confidence: 83%

A Level Set Based Shape Optimization Method for an Elliptic Obstacle Problem

Burger

Matevosyan

Wolfram

2011

Math. Models Methods Appl. Sci.

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“…without density assumptions and non-transversal intersection was proved without a sign assumption on the solution [Ind]. If F (M ) = tr(M ) this problem was investigated in [SU03,And07] and in [Mat05] when the {u = 0} term is removed, cf. [CSS04].…”

Section: Introductionmentioning

confidence: 99%

Non-transversal intersection of the free and fixed boundary in the mean-field theory of superconductivity

Indrei

2019

Interfaces Free Bound.

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Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions

Indrei

Minne

2016

Anal. PDE

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In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained such as BMOestimates, C 1,1 regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.

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Tangential Touch Between Free and Fixed Boundaries in a Problem from Superconductivity

Cited by 3 publications

References 13 publications

A Level Set Based Shape Optimization Method for an Elliptic Obstacle Problem

A Level Set Based Shape Optimization Method for an Elliptic Obstacle Problem

Non-transversal intersection of the free and fixed boundary in the mean-field theory of superconductivity

Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions

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