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We study the scaling behaviour of a class of compatible two-well problems forhigher order, homogeneous linear differential operators. To this end, we firstdeduce general lower scaling bounds which are determined by the vanishing orderof the symbol of the operator on the unit sphere in direction of the associatedelement in the wave cone. We complement the lower bound estimates by a detailedanalysis of the two-well problem for generalized (tensor-valued) symmetrizedderivatives with the help of the (tensor-valued) Saint-Venant compatibilityconditions. In two spatial dimensions for highly symmetric boundary data (butarbitrary tensor order $m \in \mathbb{N}$) we provide upper bound constructionsmatching the lower bound estimates. This illustrates that for the two-wellproblem for higher order operators new scaling laws emerge which are determinedby the Fourier symbol in the direction of the wave cone. The scaling for thesymmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23}provides an example of this family of new scaling laws.

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