相关论文: Jump-preserving polynomial interpolation in non-ma…

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $\Omega\subset\mathbb{R}^N$ and in $\mathbb{R}^N\setminus\Omega$ for $N\geq3$. The boundary…

A symplectic integration of a Poisson manifold $(M,\Lambda)$ is a symplectic groupoid $(\Gamma,\eta)$ which realizes the given Poisson manifold, i.e. such that the space of units $\Gamma_0$ with the induced Poisson structure $\Lambda_0$ is…

Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node…

We establish a relation between the approximation in $L^2[-\pi,\pi]$ by exponentials with the set of frequencies of Beurling--Malliavin density less than $1$ and the meromorphic interpolation at $\mathbb Z$. Furthermore, we show that…

In this paper, we will give a horizontal gradient estimate of positive solutions of $\Delta_b u = - \lambda u$ on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive…

We show that, for a constant-degree algebraic curve $\gamma$ in $\mathbb{R}^D$, every set of $n$ points on $\gamma$ spans at least $\Omega(n^{4/3})$ distinct distances, unless $\gamma$ is an {\it algebraic helix} (see Definition 1.1). This…

The solution $u$ of an elliptic interface problem in a domain $\Omega$ is often smooth away from the interface $\Gamma\subset \Omega$, but its gradient is discontinuous across $\Gamma$, resulting in low regularity; in particular, $u \notin…

We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ \Omega,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\…

Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution…

Suppose $\Omega$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=\min \left\{\sigma\geq 1: \, \Omega\subset \sigma S\right\}$. Here $\sigma S$ is the result of homothety of…

We say that a PDE in a Riemannian manifold $M$ is geometric if,$\ $whenever $u$ is a solution of the PDE on a domain $\Omega$ of $M$, the composition $u_{\phi}:=u\circ\phi$ is also solution on $\phi^{-1}\left( \Omega\right) $, for any…

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in $\mathbb{C}^d$. We look at differentiability at a boundary…

We establish the higher differentiability for the minimizers of the following non-autonomous integral functionals \begin{equation*} \mathcal{F}(u,\Omega):= \, \int_\Omega \sum_{i=1}^{n} \, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx,…

Given a residually connected incidence geometry $\Gamma$ that satisfies two conditions, denoted $(B_1)$ and $(B_2)$, we construct a new geometry $H(\Gamma)$ with properties similar to those of $\Gamma$. This new geometry $H(\Gamma)$ is…

A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $\Omega=\{(x',x_n):\,x_n>\varphi(x')\}\times\mathbb R$,…

This article studies (optimal) $W^{2m-1,\infty}$-regularity for the polyharmonic equation $(-\Delta)^m u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$, where $\Gamma$ is a (suitably regular) $(n-1)$-dimensional submanifold of $\mathbb{R}^n$,…

We show that the spaces $\mathcal{E}_{\{\omega\}}(\Omega)$ of ultradifferentiable functions of Roumieu type satisfy the dual interpolation estimate for small theta, where $\omega$ is a quasianalytic weight function and $\Omega$ is an…

We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbf{\Pi}$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this…

We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subset of primes. These inequalities provide endpoints for the $r$-variational estimates obtained by Trojan…

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u)&=\mu^{+}_{1}(\Omega)u &&\text{in }\Omega,\\ u &=0 &&\text{on }\partial\Omega,…