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Abstract In this paper, we present counterexamples to maximal$$L^p$$ -regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal$$L^2$$ -regularity on$$H^{-1}$$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal$$L^p$$ -regularity on$$H^{-1}(\mathbb {R}^d)$$ or$$L^2$$ -regularity on$$L^2(\mathbb {R}^d)$$ .more » « less
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Abstract We construct nonlinear entire anisotropic minimal graphs over$$\mathbb{R}^{4}$$ , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over$$\mathbb{R}^{n},\, n \geq 8$$ .more » « less
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We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.more » « less
