Related papers: A Self-dual Polar Factorization for Vector Fields

Reconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e., the…

A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in $\mathbb{R}^3$, decaying at infinity, from the knowledge of its $\nabla \times A$ and $\nabla \cdot A$. The representation…

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $\Omega$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= \Delta n - \chi \nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n -…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem $$\mathcal{L}_\varepsilon^s u_\varepsilon =(-\nabla\cdot (A_\varepsilon(x)\nabla))^{s}u_\varepsilon=f \mbox{ in } \mathcal O, $$ with…

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, we define and prove the existence of a partition of…

Following two different tracks, we arrive at a definition of Nahm's transformation valid for self-dual fields on the 4-dimensional torus with non-zero twist tensor.The transform is again a self-dual gauge field defined on a new torus and…

We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures $\mu,\nu$ ordered with respect to a cone $\mathcal{F}$ of functions on $\Omega$ stable under…

One studies here, via the La Salle invariance principle for nonlinear semigroups in Banach spaces, the properties of the $\omega$-limit set $\omega(u_0)$ corresponding to the orbit $\gamma(u_0)=\{u(t,u_0);\ t\ge0\}$, where $u=u(t,u_0)$ is…

A real matrix is Hurwitz if its eigenvalues have negative real parts. The following generalisation of the Bidimensional Global Asymptotic Stability Problem (BGAS) is provided: Let $X:R^2-->R^2$ be a C^1 vector field whose derivative DX(p)…

Let $k$ be a field and let $\Lambda$ be an indecomposable finite dimensional $k$-algebra such that there is a stable equivalence of Morita type between $\Lambda$ and a self-injective split basic Nakayama algebra over $k$. We show that every…

Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}),…

We show that the unitary factor U in the polar decomposition of a nonsingular matrix Z = U H is the minimizer for both ||Log(Q^* Z)|| and ||sym (Log(Q^*Z))|| over unitary Q, for any given invertible complex n-times-n matrix Z, for any…

In this paper we introduce a new logarithmic double phase type operator of the form\begin{align*}\mathcal{G}u:=-\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u+\mu(x)\left[\log(e+|\nabla u|)+\frac{|\nabla u|}{q(x)(e+|\nabla…

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite…

For any smooth bounded domain $\Omega \subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(\Omega)$ for all $p < \infty$, and magnetic fields $B^\epsilon \in L_t^p C^{m}(\Omega)$ for all $p < \infty$ and…

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot \sigma \nabla)^s u = 0 \quad \text{in } \Omega \subset \mathbb{R}^n, \] where $0 < s < 1$, $n…

In a previous paper we presented the renormalization of Einstein-Hilbert gravity under inclusion of higher derivative terms and proposed a projection down to the physical state space of Einstein-Hilbert. In the present paper we describe…

In this paper we construct nontrivial exterior domains $\Omega \subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, {1mm] \ u= 0 & \mbox{on }\; \partial \Omega,…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…