Radial symmetry of p-harmonic minimizers

TL;DR

This paper proves that planar radial minimizers for the p-harmonic energy are globally minimal under boundary conditions, but not necessarily in the traction free setting, addressing a longstanding open question in nonlinear elasticity.

Contribution

It establishes the global minimality of radial p-harmonic minimizers under boundary conditions, clarifying their role in nonlinear elasticity.

Findings

Radial minimizers are globally minimal with prescribed boundary data.

Identity map may not be a global minimizer in traction free conditions.

Addresses open questions about cavitating minimizers in nonlinear elasticity.

Abstract

"It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557--611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy". The quotation is from [J. Sivaloganathan and S. J. Spector, Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity, Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008), no. 1, 201--213] and seems to be still accurate. The model case of the $p$-harmonic energy is considered here. We prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary. In the traction free setting, however, even the identity map need not be a global minimizer.