Abstract
We consider the space of functions of bounded -variation. For a given first-order linear homogeneous differential operator with constant coefficients , this is the space of -functions such that the distributional differential expression is a finite (vectorial) Radon measure. We show that for Lipschitz domains , -functions have an -trace if and only if is -elliptic (or, equivalently, if the kernel of is finite-dimensional). The existence of an -trace was previously only known for the special cases that coincides either with the full or the symmetric gradient of the function (and hence covered the special cases BV or BD). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the BV- and BD-settings) but rather compare projections onto the nullspace of as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on .
Citation
Dominic Breit. Lars Diening. Franz Gmeineder. "On the trace operator for functions of bounded $\mathbb{A}$-variation." Anal. PDE 13 (2) 559 - 594, 2020. https://doi.org/10.2140/apde.2020.13.559
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