Gamma-convergence of discrete approximations to interfaces with prescribed mean curvature (original) (raw)
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Mathematical Models and Methods in Applied Sciences, 1999
In this paper we consider the so-called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend Ref. 10 with the introduction of the same nonregular potential used for the evolution problem in Ref. 9 and prove an optimal [Formula: see text] error estimate for nondegenerate minimizers (where ε represents the perturbation parameter). Following Ref. 10 the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error estimate for discrete minima using the Rannacher–Scott...
Regularity of minimizers for a class of anisotropic free discontinuity problems
2001
This paper contains existence and regularity results for solutions u : Ω → R n N of a class of free discontinuity problems i.e.: the energy to minimize consists of both a bulk and a surface part. The main feature of the class of problems considered here is that the energy density of the bulk part is supposed to be fully anisotropic with p-growth in the scalar case, n = 1. Similar results for the vectorial case n > 1 are obtained for radial energy densities, being anisotropic again with p-growth.
A comparison principle for minimizers
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000
We give some conditions that ensure the validity of a Comparison principle for the minimizers of integral functionals, without assuming the validity of the Euler-Lagrange equation. We deduce a weak maximum principle for (possibly) degenerate elliptic equations and, together with a generalization of the bounded slope condition, the Lipschitz continuity of minimizers. To prove the main theorem we give a result on the existence of a representative of a given Sobolev function that is absolutely continuous along the trajectories of a suitable autonomous system. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Un principe de comparaison pour les minima Résumé. Nous donnons des conditions qui assurent la validité d'un principe de comparaison pour les minimums d'une fonctionnelle intégrale qui ne satisfont pas nécessairement à l'équation d'Euler-Lagrange. Nous en déduisons un principe de maximum faible pour les équations elliptiques (éventuellement) dégénerées et, en généralisant la condition de la pente bornée, la Lipschitz continuité des minimums. La preuve du théorème principal se base sur l'éxistence d'un représentant d'une fonction de Sobolev donnée qui est absolument continu sur les trajectoires d'un système autonome convenable. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée Nous fixons un ouvert borné Ω de R n. La fonction L(x, z, p) est définie dans Ω × R × R n et est une fonction dans W 1,q (Ω), q 1. La fonctionū est dans W 1,q (Ω) et on pose W 1,q u (Ω) =ū + W 1,q 0 (Ω). Dans cette partie nous nous référons aux hypothèses A, A , B et D du texte anglais qui suit. THÉORÈME PRINCIPAL 1 ([4]).-On suppose que (L,) satisfait à l'hypothèse B. Soit w un minimum de I(u) = Ω L x, u(x), ∇u(x) dx dans W 1,q u (Ω). Si w dans ∂Ω, alors w presque partout dans Ω. Note présentée par Haïm BRÉZIS.
Annales de l'Institut Henri Poincare (C) Non Linear Analysis
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General Methods of Elliptic Minimization
arXiv (Cornell University), 2016
We provide general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. Given a collection of competing "surfaces," which span a given "bounding set" in an ambient metric space, we produce one minimizing an elliptic area functional. The collection of competing surfaces is assumed to satisfy a set of geometrically-defined axioms. These axioms hold for collections defined using any combination of homological, cohomological or linking number spanning conditions. A variety of minimization problems can be solved, including sliding boundaries.
V.: On the regularity of critical and minimal sets of a free interface problem
2014
We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin [13].
In the present paper we consider a numerical method for Variational Fracture based on classical finite elements with gaps, the novelty being the way in which the FE mesh is moved to approximate the cracks. Indeed VF requires the ability to locate and approximate the crack fronts. On adopting the dicrete, "strong discontinuity" approach cracks cannot be restricted to the skeleton of a fixed FE mesh. With our method mesh is made variable: mesh nodes are taken as further unknowns and the minimization of the Energy is considered, at the same time, with respect to displacements and positions of the nodes of the mesh in the reference configuration (minimization over variable triangulations). The results we present are still of a research type. To simplify the computations, the mesh we adopt are still too coarse to accurately catch the large values of the gradient of displacement that arise at the crack tips and so the energy. Also the descent strategy we use to solve the variational problem, though the most natural, is numerically rather slow. The issue we try to address here is to verify the possibility of tracking cracks by using FE with gaps, without resorting to sophisticated numerical tools, by working on simple benchmark problems. SOMMARIO. In questo lavoro si presenta un metodo numerico per l'approssimazione di problemi di Frattura variazionale, basato su elementi finiti classici con discontinuità finite. La novità del metodo risiede nel modo in cui la mesh si adatta per approssimare le fratture. Adottando l'approccio discreto, le fratture non possono essere ristrette allo scheletro di una mesh fissa. Con il nostro metodo la mesh è resa variabile (ed adattabile) considerando la posizione dei nodi della mesh come un'ulteriore variabile, la minimizzazione dell'energia si effettua allo stesso tempo rispetto alla variazione degli spostamenti e delle coordinate dei nodi della mesh nella configurazione di riferimento (minimizzazione su triangolazioni variabili). I risultati che presentiamo sono ancora preliminari, in quanto per contenere i tempi di computazione, la mesh che si adotta è piuttosto rada per essere in grado di rappresentare accuratamente i forti gradienti che sono presenti all'apice della frattura e, di conseguenza, l'energia del sistema. Anche la strategia di discesa che adottiamo per ottenere le soluzioni di minimo, sebbene la più naturale, è notoriamente lenta. L'obiettivo qui è mostrare la possibilità di individuare e tracciare le fratture utilizzando elementi finiti con discontinuità forti, senza ricorrere a strumenti numerici sofisticati e lavorando su esempi di complessità limitata.