Related papers: Regularity for graphs with bounded anisotropic mea…

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a…

We prove the low Mach number limit of non-isentropic ideal magnetohydrodynamic (MHD) equations with general initial data in the half-space whose boundary satisfies the perfectly conducting wall condition. By observing a special structure…

We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in $\mathbb{R}^n$. Namely we study \begin{equation*} \left\{\begin{array}{l }…

We prove that the linearly controlled asymptotic dimension of the fundamental group of any 3-dimensional graph-manifold does not exceed 7. As applications we obtain that the universal cover of such a graph-manifold is an absolute Lipschitz…

We study the motion of sets by anisotropic curvature under a volume constraint in the plane. We establish the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of Wulff shapes of equal area, the…

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the…

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…

Suppose $ F $ is an integrand associated with a uniformly convex $ \mathscr{C}^{3} $-norm, and $ V $ is a $ n $-dimensional varifold in an open subset of $ \mathbf{R}^{n+1} $ such that $ \mathscr{H}^n \llcorner \operatorname{spt} \| V \| $…

We study local regularity properties of solutions to stationary anisotropic magnetic Schr\"odinger equations in $\mathbb{R}^d$, $d \ge 2$, arising from singular magnetic potentials concentrated along manifolds of general codimension $2 \le…

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature…

We obtain necessary conditions for the existence of complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. In the two-dimensional case we prove Bernstein-type results in each of these ambient spaces.

We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of…

In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where $\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is…

This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework…

In this paper, we study the finite boundary regularity and estimates of an asymptotically hyperbolic Einstein manifold in even dimension $n+1.$ We show that if the initial compactification is $C^{n-1}$ and the $(n-3)$-th derivative of its…

We construct 2-surfaces of prescribed mean curvature in 3-manifolds carrying asymptotically flat initial data for an isolated gravitating sysqtem with rather general decay conditions. The surfaces in question form a regular foliation of the…

In the present article we prove second-order and Lipschitz regularity for quasilinear elliptic equations in metric spaces endowed with a lower bound on the Ricci curvature. The estimates we obtain are quantitative and cover a large class of…

In this paper we consider Riemannian manifolds of dimension at least $3$, with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski…

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the…

We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…