A dynamical approach to the variational inequality on modified elastic graphs

Shinya Okabe,Kensuke Yoshizawa

doi:10.1515/geofl-2020-0100

Abstract

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.

Highlights

In this paper we are interested in a dynamical approach to the obstacle problem for modi ed elastic graphs: nd u : [, ] → R such that min v∈Kψ Eλ (v) (1.1)with Eλ(v) := λL(v) + W(v), where λ is a given nonnegative constant, L(v) := (v (x)) dx = + v (x) dx, (1.2) W(v) :=κv(x) (v (x)) dx = (v (x) + v (x) ) / dx
We consider the variational inequality on modi ed elastic graphs
Since the variational inequality is derived from the minimization problem for the modi ed elastic energy de ned on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations

Summary

Introduction

In this paper we are interested in a dynamical approach to the obstacle problem for modi ed elastic graphs: nd u : [ , ] → R such that min v∈Kψ. Problem (1.1) can be solved by the direct method of calculus of variations, it is signi cant to prove the existence of solutions to (1.6) via the other strategy. Because, it is not clear whether the set of all minimizers of (1.1) is equivalent to the set of all solutions to (1.6) or not, due to the lack of the convexity of the functional λL + W. The solution u subconverges to a solution u* ∈ Kψ of (1.6) as t → ∞

Shinya Okabe and Kensuke Yoshizawa

Approximate solutions

The inequality is reduced into

Findings

This together with the Schwartz inequality implies that x