Related papers: Minimally singular functions and the rigidity prob…

We consider a generalization of the Cheeger problem in a bounded, open set $\Omega$ by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any…

Let $M$ be a compact complex manifold of dimension $n\geq 2$. We prove that for any Hermitian metric $\omega$ on $M$, there exists a unique smooth function $f$ (up to additive constants) such that the conformal metric $\omega_g =e^f \omega$…

The Eremenko-Lyubich class consists of transcendental entire functions with bounded singular set and the Speiser class is made up of functions with a finite singular set. In an earlier paper "Models for the Eremenko-Lyubich class" I gave a…

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a…

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. When we consider sub-Riemannian manifolds, our definition coincide with…

We study functions of bounded variation (and sets of finite perimeter) on a convex open set $\Omega\subseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an…

Let $\Omega$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $\Sigma_k$ be a closed smooth submanifold of $\delta \Omega$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight…

We prove that every smooth closed manifold admits a smooth real-valued function with only two critical values. We call a function of this type a \emph{Reeb function}. We prove that for a Reeb function we can prescribe the set of minima (or…

If a smooth function of one variable has maximum one on the unit interval, and has there $d$ zeroes, then its $(d+1)$-st derivative must be "big". This is one of the simplest examples of what we call "smooth rigidity": certain geometric…

We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $\mathrm{BD}(\Omega)$ have weak gradients in $\mathrm{L}_{\mathrm{loc}}^{1}(\Omega;\mathbb{R}^{n\times n})$. This is achieved for…

We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $\Lambda$ and when the surfaces are…

A function $f:\RR \to \RR$ is called \emph{vertically rigid} if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. We prove Jankovi\'c's conjecture by showing that a continuous function is vertically rigid if and only if it is of…

In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming…

We establish certain fine properties for functions of bounded $\mathscr A$-variation known in the classical $BV$ setting. Here, $\mathscr A$ is a $k$th order constant-coefficient homogeneous linear differential operator with a…

We consider functions $\omega$ on the unit circle $\mathbb T$ with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions and find an asymptotic formula for the distance in the BMO-norm…

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous…

For a given domain $\Omega \subset \Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $\Omega$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower…

In this paper we study the right differentiability of a parametric infimum function over a parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to…

Tempered Exponential Measures (TEMs) are a parametric generalization of the exponential family of distributions maximizing the tempered entropy function among positive measures subject to a probability normalization of their power…

We prove uniqueness for minimizers of the weighted least gradient problem \[\inf \left\lbrace \int_{\Omega} a|Du|: \ \ u\in BV(\Omega), \ \ u|_{\partial \Omega}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous and it…