We establish that trace inequalities for vector fields u∈ C c∞ ℝ n, ℝ N (*) D k− 1 u L n− s/n−
1 d μ⩽ c μ L 1, n− sn− 1/n− s AD u L 1 d L n hold if and only if the k-th order homogeneous …

We discuss practical methods for computing the space of solutions to an arbitrary
homogeneous linear system of partial differential equations with constant coefficients. These …

We introduce new spaces of martingales that behave like L1-based Sobolev spaces on Rd.
We prove a martingale analog of Van Schaftingen's theorem and give sharp estimates on …

In this paper we settle Iwaniec and Sbordone's 1994 conjecture concerning very weak
solutions to the p-Laplace equation. Namely, on the one hand we show that distributional …

We consider the energy critical semilinear heat equation $$\left\{\begin {aligned} &\partial_t
u-\Delta u=| u|^{\frac {4}{n-2}} u &\mbox {in}{\mathbb R}^ n\times (0, T),\\&u (x, 0)= u_0 …

Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric
measure theory. The last four decades have seen the emergence of a wealth of connections …

We prove that m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in
L p, p> m, are regular almost everywhere in every dimension and codimension. This …

Let $\mathcal {W} $ be a closed dilation and translation invariant subspace of the space of
$\mathbb {R}^\ell $-valued Schwartz distributions in $ d $ variables. We show that if the …

In this article we study quantitative rigidity properties for the compatible and incompatible
two-state problems for suitable classes of A-free differential inclusions and for a singularly …

We give two characterizations, one for the class of generalized Young measures generated
by\, A\, A-free measures and one for the class generated by B B-gradient measures B u B u …