On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators

TL;DR

This paper investigates the scaling behavior of two-well problems for higher order linear differential operators, establishing bounds and revealing new scaling laws influenced by the Fourier symbol and wave cone geometry.

Contribution

It derives general lower bounds for scaling, analyzes specific two-well problems with symmetry, and identifies novel scaling laws for higher order operators based on the Fourier symbol.

Findings

Lower scaling bounds depend on the symbol's vanishing order.

Explicit upper bounds are constructed for symmetric boundary data.

New scaling laws emerge for higher order operators influenced by the wave cone.

Abstract

We study the scaling behaviour of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order $m \in \mathbb{N}$) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23} provides an example of this family of new scaling laws.