Search results for "Critical set"

showing 4 items of 4 documents

Remark on a nonlocal isoperimetric problem

2017

Abstract We consider isoperimetric problem with a nonlocal repulsive term given by the Newtonian potential. We prove that regular critical sets of the functional are analytic. This optimal regularity holds also for critical sets of the Ohta–Kawasaki functional. We also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set.

Newtonian potentialcritical pointsApplied Mathematics010102 general mathematicsMathematical analysista111Isoperimetric dimension01 natural sciences010101 applied mathematicsMathematics - Analysis of PDEsshape optimizationFOS: Mathematicsisoperimetric problemShape optimizationBall (mathematics)0101 mathematicsIsoperimetric inequalityAnalysisCritical setAnalysis of PDEs (math.AP)MathematicsNonlinear Analysis: Theory, Methods and Applications
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Analysis of a slow–fast system near a cusp singularity

2016

This paper studies a slow fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results. (C) 2015 Elsevier Inc. All rights reserved.

Cusp (singularity)0209 industrial biotechnologyDifferential equationApplied Mathematics010102 general mathematicsMathematical analysis[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS]02 engineering and technologyDynamical Systems (math.DS)01 natural sciencesPerturbation-theory020901 industrial engineering & automationSlow manifoldNormal form theoryFOS: MathematicsDifferential-equationsPerturbation theory (quantum mechanics)0101 mathematicsMathematics - Dynamical SystemsAnalysisCritical setMathematics
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Stationary sets of the mean curvature flow with a forcing term

2020

We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space $\mathbb R^n$ of dimension at least 2. Our main results states that tangential balls in $\mathbb R^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in $\mathbb R^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.

osittaisdifferentiaaliyhtälötMean curvature flowForcing (recursion theory)Mean curvatureEuclidean spaceApplied Mathematics010102 general mathematicsMathematical analysisstationary setscritical setsvariaatiolaskenta01 natural sciences35J93Term (time)010101 applied mathematicsMathematics - Analysis of PDEsFlow (mathematics)forced mean curvature flowBounded functionFOS: Mathematics0101 mathematicsConstant (mathematics)AnalysisAnalysis of PDEs (math.AP)MathematicsAdvances in Calculus of Variations
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Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane

2020

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent. peerReviewed

osittaisdifferentiaaliyhtälötMathematics - Analysis of PDEsforced mean curvature flowFOS: Mathematicsstationary setscritical setsGeometry and TopologyAstrophysics::Earth and Planetary Astrophysicslarge time behaviorAnalysis of PDEs (math.AP)
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