TY - JOUR

T1 - The Dirichlet Principle for Inner Variations

AU - Iwaniec, Tadeusz

AU - Onninen, Jani

N1 - Publisher Copyright:
Copyright © 2020, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - We are concerned with the Dirichlet energy of mappings defined on domains in the complex plane. The motivation behind our questions, however, comes from more general energy integrals of mathematical models of Hyperelasticity. The Dirichlet Principle, the name coined by Riemann, tells us that the outer variation of a harmonic mapping increases its energy. Surprisingly, when one jumps into details about inner variations, which are just a change of independent variables, new equations and related questions start to matter. The inner variational equation, called the Hopf Laplace equation, is no longer the Laplace equation. Its solutions are generally not harmonic; we refer to them as Hopf harmonics. The natural question that arises is how does a change of variables in the domain of a Hopf harmonic map affect its energy? We show, among other results, that in case of a simply connected domain the energy increases. This should be viewed as Riemann’s Dirichlet Principle for Hopf harmonics. The Dirichlet Principle for Hopf harmonics in domains of higher connectivity is not completely solved. What complicates the matter is the insufficient knowledge of global structure of trajectories of the associated Hopf quadratic differentials, mainly because of the presence of recurrent trajectories. Nevertheless, we have established the Dirichlet Principle whenever the Hopf differential admits closed trajectories and crosscuts. Regardless of these assumptions, we established the so-called Infinitesimal Dirichlet Principle for all domains and all Hopf harmonics. Precisely, the second order term of inner variation of a Hopf harmonic map is always nonnegative. The topics presented in this paper open new directions toward mathematical foundations of Hyperelasticity. In particular, the use of quadratic differentials in the context of hyperelasticity should appeal to both mathematical analysts and researchers in the engineering fields.MSC Codes 31A05, 30G20, 35J25

AB - We are concerned with the Dirichlet energy of mappings defined on domains in the complex plane. The motivation behind our questions, however, comes from more general energy integrals of mathematical models of Hyperelasticity. The Dirichlet Principle, the name coined by Riemann, tells us that the outer variation of a harmonic mapping increases its energy. Surprisingly, when one jumps into details about inner variations, which are just a change of independent variables, new equations and related questions start to matter. The inner variational equation, called the Hopf Laplace equation, is no longer the Laplace equation. Its solutions are generally not harmonic; we refer to them as Hopf harmonics. The natural question that arises is how does a change of variables in the domain of a Hopf harmonic map affect its energy? We show, among other results, that in case of a simply connected domain the energy increases. This should be viewed as Riemann’s Dirichlet Principle for Hopf harmonics. The Dirichlet Principle for Hopf harmonics in domains of higher connectivity is not completely solved. What complicates the matter is the insufficient knowledge of global structure of trajectories of the associated Hopf quadratic differentials, mainly because of the presence of recurrent trajectories. Nevertheless, we have established the Dirichlet Principle whenever the Hopf differential admits closed trajectories and crosscuts. Regardless of these assumptions, we established the so-called Infinitesimal Dirichlet Principle for all domains and all Hopf harmonics. Precisely, the second order term of inner variation of a Hopf harmonic map is always nonnegative. The topics presented in this paper open new directions toward mathematical foundations of Hyperelasticity. In particular, the use of quadratic differentials in the context of hyperelasticity should appeal to both mathematical analysts and researchers in the engineering fields.MSC Codes 31A05, 30G20, 35J25

KW - Harmonic mappings

KW - Holomorphic quadratic differentials

KW - Hopf-Laplace equation

KW - Monotone mappings

KW - The principle of non-interpenetration of matter

UR - http://www.scopus.com/inward/record.url?scp=85094548523&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85094548523&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85094548523

JO - Nuclear Physics A

JF - Nuclear Physics A

SN - 0375-9474

ER -