相关论文: Fuglede conjecture holds for convex planar domains

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

The bounded orbit conjecture says that every homeomorphism on the plane with each of its orbits being bounded must have a fixed point. Brouwer's translation theorem asserts that the conjecture is true for orientation preserving…

For any pseudoconvex Runge domain $\Omega\subset\mathbb{C}^2$ we prove that every closed discrete subset in $\Omega$ is contained in a properly embedded complex curve in $\Omega$ with any prescribed topology (possibly infinite).

The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling…

In this note, we show that the strong Viterbo conjecture holds true on any convex toric domain, and that the Viterbo's volume-capacity conjecture holds for the product of a $1$-unconditional convex body $A\subset\mathbb{R}^{n}$ and its…

Using recent developments in the theory of mixed motives, we prove that the log Bloch conjecture holds for an open smooth complex surface if the Bloch conjecture holds for its compactification. This verifies the log Bloch conjecture for all…

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of…

Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial…

We consider shape functionals of the form $F_q(\Omega)=P(\Omega)T^q(\Omega)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(\Omega)$ denotes the perimeter of $\Omega$ and $T(\Omega)$ is the torsional…

We prove a necessary optimality condition of Euler--Lagrange type for the calculus of variations with Omega derivatives, which turns out to be sufficient under jointly convexity of the Lagrangian.

We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb…

Let $\Omega$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that the trace of the space $C^2(\Omega)$ to the boundary of $\Omega$ has the following finiteness property: A function $f:\partial\Omega\to R$ is the trace to the…

We construct a bounded domain $\Omega$ in $\mathbb{C}^2$ with boundary of class $\mathcal{C}^{1,1}$, such that $\overline{\Omega}$ has a Stein neighborhood basis, but is not $s$-H-convex for any real number $s\geq{1}$.

Let \Omega and \tilde{\Omega} be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: \Omega \to \tilde{\Omega} such that the graph \Sigma = \{(x,f(x)): x \in \Omega\} is a minimal…

Let $\Omega \subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(\Omega)^2$ for every $f\in L^p_0(\Omega)$, if…

Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd and let \T be a discrete subset of \Rd. We show that the exponentials $e_t(x):=exp(i2\pi tx)$, $t\in\T$ form an othonormal basis for $L^2(\Q)$ if and only if the…

Let $\hat \Omega \subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $\hat \Omega$. Starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial \hat…

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,…

We study an elliptic system coupled by Monge-Amp\`{e}re equations: \begin{center} $\left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^\alpha, & \hbox{in $\Omega,$} det~D^{2}u_{2}={(-u_{1})}^\beta, & \hbox{in $\Omega,$} u_{1}<0, u_{2}<0,&…