相关论文: Tur\'an's extremal problem for positive definite f…

The following problem originated from a question due to Paul Turan. Suppose $\Omega$ is a convex body in Euclidean space $\RR^d$ or in $\TT^d$, which is symmetric about the origin. Over all positive definite functions supported in $\Omega$,…

We study the following question posed by Turan. Suppose K is a convex body in Euclidean space which is symmetric with respect to the origin. Of all positive definite functions supported in K, and with value 1 at the origin, which one has…

Let G be a locally compact abelian group (LCA group) and U be an open, 0-symmetric set. Let F:=F(U) be the set of all real valued continuous functions from G to R which are supported in U and are positive definite. The Turan constant T(U)…

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $\text{Re}…

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ ($n \ge 2$). In this work, we prove that if there exists an integrable function $f$ such that it's Radon transform over $(n-1)$-dimensional hyperplanes intersecting the domain…

In the general setting of a locally compact Abelian group $G$, the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions $f: G \to \mathbb{R}$ satisfying $f(0) = 1$ and…

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(\Omega)/(M(\Omega)|\Omega|)$ and $M(\Omega)\lambda_1(\Omega) $, where $\Omega$ is a…

One of the classical problems concerns the class of analytic functions $f$ on the open unit disk $|z|<1$ which have finite Dirichlet integral $\Delta(1,f)$, where $$\Delta(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1). $$ The class…

Let $\Omega$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $\Omega$ that vanishes on the boundary of $\Omega$ is called a Martin function. In this note, we show that, when $\Omega$ is…

We study the problem of finding a function $f$ with ``small support'' that simultaneously tiles with finitely many lattices $\Lambda_1, \ldots, \Lambda_N$ in $d$-dimensional Euclidean spaces. We prove several results, both upper bounds…

In this paper, we investigate the compactness of extremal functions for a critical singular anisotropic Trudinger-Moser inequality established by Lu-Shen-Xue-Zhu\cite{ref1}. We prove by means of blow-up analysis that the extremals…

Let $\ell>0$ be arbitrary. We introduce the extremal quantities $$ G(\ell):=\frac{\sup_{f} \int_{-\ell}^{\ell} f\,dx}{\int_{-1}^1 f\,dx},\quad C(\ell):=\frac{\sup_{f} \sup_{a\in {\mathbb R}} \int_{a-\ell}^{a+\ell} f\,dx}{\int_{-1}^1 f\,dx},…

We consider the relaxation of polyconvex functionals with linear growth with respect to the strict convergence in the space of functions of bounded variation. These functionals appears as relaxation of $F(u,\Omega):=\int_\Omega f(\nabla…

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain that can be written as $\Omega=\bigcup_{t} \Omega_t$, where $\{\Omega_t\}_{t\in\Gamma}$ is a countable collection of domains with certain properties. In this work, we develop a technique…

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…

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 $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,\Omega\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (\Omega)\,$ (known as {\em torsional…

For a bounded domain $\Omega\subset \mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\|u\|^p_{\infty}\le \int_\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\Omega)$. We show…

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal…