相关论文: Fuglede conjecture holds for convex planar domains

We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial \Omega \setminus F$ is $\mathcal C^{2}$-smooth and locally…

Let $\Omega\subset\mathbb{C}$ be a bounded domain such that there exists an algebraic harmonic function of degree two vanishing on the boundary of $\Omega.$ Then we show that the Khavinson-Shapiro conjecture holds for $\Omega:$ if the…

Let $\Phi$ : R n $\rightarrow$ R $\cup$ {+$\infty$} be an even convex function and L$\Phi$ be its Legendre transform. We prove the functional form of Mahler conjecture concerning the functional volume product P ($\Phi$) = e --$\Phi$ e…

Let Q be a nondegenerate quadratic form, and L is a nonzero linear form of dimension d>3. As a generalization of the Oppenheim conjecture, we prove that the set {(Q(x),L(x)):x\in Z^d} is dense in R^2 provided that Q and L satisfy some…

We describe a class of measurable subsets $\Omega$ in $\br^d$ such that $L^2(\Omega)$ has an orthogonal basis of frequencies $e_\lambda(x)=e^{i2\pi\lambda\cdot x}(x\in\Omega)$ indexed by $\lambda\in\Lambda\subset\br^d$. We show that such…

We give a sharp lower bound on the area of a domain that can be enclosed by a closed embedded $\lambda$-convex curve of a given length on the Lobachevsky plane.

Let $X$ be a smooth compact complex surface subject to the following conditions: (i) the canonical line bundle $\mathcal{O}_X(K_X) $ is very ample, (ii) the irregularity $q(X): = h^1(\mathcal{O}_X) =0$, (iii) $X$ contains no rational normal…

In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…

Let Omega be a bounded, simply connected domain with boundary of class C^{1,1} except at finitely many points S_j where the boundary is locally a corner of aperture alpha_j<=pi/2. Improving on results of Grisvard, we show that the solution…

We show that if $\Omega$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\Omega$ does not contain any immersed…

In a recent article (2022) we proved with L. Zaj\'i\v{c}ek that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave…

We construct a new counterexample confirming the sharpness of the Dini-type condition for the boundary of $\Omega$. In particular, we show that for convex domains the Dini-type assumption is the necessary and sufficient condition which…

In this paper we study the following torsion problem \begin{equation*} \begin{cases} -\Delta u=1~&\mbox{in}\ \Omega,\\[1mm] u=0~&\mbox{on}\ \partial\Omega. \end{cases} \end{equation*} Let $\Omega\subset \mathbb{R}^2$ be a bounded, convex…

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in several complex variables and complex geometry. Ohsawa conjectured in 1995 that…

We prove Conjecture 4.16 of the paper [EL21] of Elagin and Lunts; namely, that a smooth projective curve of genus at least 1 over a field has diagonal dimension 2.

A locally convex space (lcs) $E$ is said to have an $\omega^{\omega}$-base if $E$ has a neighborhood base $\{U_{\alpha}:\alpha\in\omega^\omega\}$ at zero such that $U_{\beta}\subseteq U_{\alpha}$ for all $\alpha\leq\beta$. The class of lcs…

We consider the heat equation in a smooth bounded convex domain $\Omega \subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_\nu u = \lambda (u - u^3)$. Stable non-constant stationary solutions do not exist when…

Let $\Omega \subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in…