Related papers: Null Lagrangian Measures in subspaces, compensated…

Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole $n$-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical…

Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical…

We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the…

Self-gravitating horizonless ultra-compact objects that possess light rings have attracted the attention of physicists and mathematicians in recent years. In the present compact paper we raise the following physically interesting question:…

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-\Delta_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a…

We find a necessary and sufficient condition for a doubling metric space to carry a (1,p)-Poincare inequality. The condition involves discretizations of the metric space and Poincare inequalities on graphs.

Second-order sufficient conditions for local optimality have been playing an important role in local convergence analysis of optimization algorithms. In this paper, we demonstrate that this condition alone suffices to justify the linear…

This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical…

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$,…

In this paper, we study the problem of compressed sensing using binary measurement matrices and $\ell_1$-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to…

Massive and massless Dirac equations with Lorentz-covariant cubic nonlinearities are considered in spatial dimension $d=2,3$. Global well-posedness of the Cauchy problem for small initial data in scale-invariant Sobolev spaces and…

We prove the Riemannian curvature-dimension condition $\mathsf{RCD}(KN,N+1)$ for an $N$-warped product $B\times_f^N F$ over a one-dimensional base space $B$ with a Lipschitz function $f: B\rightarrow \mathbb R_{\geq 0}$, provided (1) $f$ is…

This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector $\vec{\mu} \in \mathbb{R}^p$. In cases where the Kolmogorov $N$-width decays fast enough,…

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,\sigma_1\,\partial_x u + m c^2 \sigma_3 u + V(x)\,u =…

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these…

This paper proposes a novel Coordinate-Descent Augmented-Lagrangian (CDAL) solver for linear, possibly parameter-varying, model predictive control (MPC) problems. At each iteration, an augmented Lagrangian (AL) subproblem is solved by…

We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on $\mathbb R^d$ with a quadratic, conservative nonlinearity $B(x,x)$ constrained to possess various properties common to…

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat…

We give necessary conditions for when a subset of $\mathbb{T}^n$ can contain the support of some non-zero RP-measure. Among other things we show that the support of a positive RP-measure cannot be contained in reflections of inverse images…

We are interested in the cubic Dirac equation in two space dimensions. We establish the small data global existence and sharp pointwise decay results for general cubic nonlinearities without additional structure. We also prove the…