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A form of Sobolev inequalities for the symmetric gradient of vector-valuedfunctions is proposed, which allows for arbitrary ground domains in $\mathbb R^n$. In the relevant inequalities, boundary regularity of domains is replacedwith information on boundary traces of trial functions. The inequalities soobtained exhibit the same exponents as in classical inequalities for the fullgradient of Sobolev functions, in regular domains. Furthermore, they involveconstants independent of the geometry of the domain, and hence yield novelresults yet for smooth domains. Our approach relies upon a pointwise estimatefor the functions in question via a Riesz potential of their symmetric gradientand an unconventional potential depending on their boundary trace.

Sobolev inequalities for the symmetric gradient in arbitrary domains