Related papers: The linearized translator equation and application…

We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $\Lambda \subset \mathbb{R}$, if we have $\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the main results, and prove…

We introduce and study the class of linear transfers between probability distributions and the dual class of Kantorovich operators between function spaces. Linear transfers can be seen as an extension of convex lower semi-continuous…

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding…

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new…

Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite measure $\mu$ has a positive lower bound. Motivated by wide interest in the current literature, in this paper we study the existence of classical solutions for a class…

We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\infty$…

In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of…

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by…

We introduce and study the finite-approximate solvability of operator equations \(Lu = h\) in a Hilbert space setting, where a bounded operator \(L \colon U \to H\) is paired with a finite-dimensional constraint operator \(\pi \colon H \to…

We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive…

We consider elliptic transmission problems with complex coefficients across an interface. Under proper transmission conditions, that extend known conditions for well-posedness, and sub-ellipticity we derive microlocal and local Carleman…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

We consider elliptic equations of order $2m$ in a bounded domain $Q\subset\mathbb R^n$ with nonlocal boundary-value conditions connecting the values of a solution and its derivatives on $(n-1)$-dimensional smooth manifolds $\Gamma_i$ with…

Let $\phi: \R^d \longrightarrow \C$ be a compactly supported function which satisfies a refinement equation of the form $\phi(x) = \sum_{k\in\Lambda} c_k \phi(Ax - k),\quad c_k\in\C$, where $\Gamma\subset\R^d$ is a lattice, $\Lambda$ is a…

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators $$\mathcal{L}v(x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle+Df(v_{\star}(x))v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2$$ where…

A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if $\phi_k$ is a sequence of eigenfunctions of a second order differential operator on the interval $I \subset \mathbb{R}$, then any linear…

Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$. Let $\{\lambda_k\}_1^\infty$ and $\{\alpha_k\}_1^\infty$ be the sequences of the eigenvalues…

We prove, in all dimensions $n\geq 2$, that there exists a convex translator lying in a slab of width $\pi\sec\theta$ in $\mathbb{R}^{n+1}$ (and in no smaller slab) if and only if $\theta\in[0,\frac{\pi}{2}]$. We also obtain convexity and…

We study the approximation of nonlinear operators between function spaces by transformers. Our approach is to lift functions to measures supported on their graphs and leverage a recently introduced measure-theoretic view of transformers. A…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of…