A dynamical approach to the variational inequality on modified elastic graphs

2020 ◽  
Vol 5 (1) ◽  
pp. 78-101
Author(s):  
Shinya Okabe ◽  
Kensuke Yoshizawa
Keyword(s):  
Variational Inequality ◽  
Calculus Of Variations ◽  
Minimization Problem ◽  
Elastic Energy ◽  
Existence Of Solutions ◽  
Direct Method ◽  
Gradient Flow ◽  
Unilateral Constraint ◽  
Dynamical Approach

AbstractWe consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

scholarly journals Connected perimeter of planar sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Dayrens ◽  
Simon Masnou ◽  
Matteo Novaga ◽  
Marco Pozzetta
Keyword(s):  
Minimization Problem ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Representation Formula ◽  
Steiner Trees ◽  
Lower Semicontinuous Envelope ◽  
Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

Isoperimetric Problems in the Calculus of Variations

1952 ◽  
Vol 4 ◽  
pp. 257-280 ◽  
Author(s):  
William Karush
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Indirect Method ◽  
Direct Method ◽  
Isoperimetric Problems ◽  
End Points ◽  
Sufficiency Theorems ◽  
Theory Of Fields ◽  
A Direct Method ◽  
Problem Of Bolza

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

The elastic flow of curves on the sphere

2018 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Anna Dall’Acqua ◽  
Tim Laux ◽  
Lin ◽  
Paola Pozzi ◽  
Adrian Spener
Keyword(s):  
Critical Points ◽  
Elastic Energy ◽  
Gradient Flow ◽  
Closed Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Short Time ◽  
Time Existence

Abstract We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

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