Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ Omega $ that is either a bounded domain in $ mathbb{R}^{d} $ or $ mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.
Let us consider continuous minimizers $ u : bar Omega subset mathbb{R}^n to mathbb{R}^n $ of
with $ p > 1 $ and $ r > 0 $; then it is known that every component $ u^alpha $ of $ u = (u^1, ..., u^n) $ enjoys maximum principle: the set of interior points $ x $, for which the value $ u^alpha(x) $ is greater than the supremum on the boundary, has null measure, that is, $ mathcal{L}^n({ x in Omega: u^alpha (x) > sup_{partial Omega} u^alpha }) = 0 $. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case
with $ p > 1 $ and $ r > 0 $. Indeed, for a suitable boundary value, the set of the interior points $ x $, for which the value $ u^alpha(x) $ is greater than the supremum on the boundary, has a positive measure, that is $ mathcal{L}^n({ x in Omega: u^alpha (x) > sup_{partial Omega} u^alpha }) > 0 $. In this paper we show that the measure of the image of these bad points is zero, that is $ mathcal{L}^n(u({ x in Omega: u^alpha (x) > sup_{partial Omega} u^alpha })) = 0 $, provided $ p > n $. This is a particular case of a more general theorem.
We study partial Hölder regularity for nonlinear elliptic systems in divergence form with double-phase growth, modeling double-phase non-Newtonian fluids in the stationary case.
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