Related papers: Poincar\'e trace inequalities in $BV(\mathbb B^n)$…
We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and…
With direct and simple proofs, we establish Poincar\'{e} type inequalities (including Poincar\'{e} inequalities, weak Poincar\'{e} inequalities and super Poincar\'{e} inequalities), entropy inequalities and Beckner-type inequalities for…
The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider the…
We study a variational functional of Trudinger-Moser type associated with one-sided Borel probability measure. Its boundedness at the extremal parameter holds when the residual vanishing occurs. In the proof we use a variant of the Y.Y. Li…
In the setting of a metric space equipped with a doubling measure supporting a $(1,1)$-Poincar\'e inequality, we study the problem of minimizing the BV-energy in a bounded domain $\Omega$ of functions bounded between two obstacle functions…
In the setting of a metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we define and study a class of BV functions with zero boundary values. In particular, we show that the class is the closure of…
We prove the existence of minimizers for some constrained variational problems on $BV(\Omega)$, under subcritical and critical restrictions, involving the affine energy introduced by Zhang in \cite{Z}. Related functionals have non-coercive…
For any convex set $\Omega \subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincar\'e constant in $W ^ {1, 1}(\Omega)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an…
The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by the first and last coordinates are applied to functions belonging to quasi-homogeneous, mixed-norm…
The central aim of this paper is to study (regional) fractional Poincar\'e type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various…
We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincar\'e-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is…
We investigate the character of the linear constraints which are needed for Poincar\'e and Korn type inequalities to hold. We especially analyze constraints which depend on restriction on subsets of positive measure and on the trace on a…
In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that any set of finite perimeter can be approximated in the BV norm by a set whose topological and measure theoretic boundaries…
We develop a general theory for the existence of extremal K\"ahler metrics of Poincar\'e type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we…
We give an exact formula for the Bellman function of the weak type of martingale transform. We also give the extremal functions (actually extremal sequences of functions). We find them using the precise form of the Bellman function. The…
In the paper we investigate Trudinger-Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of estremal functions or blow-up, where the domain is the ball…
Under Poincar\'e-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional…
We present a Korn-Poincar\'e-type inequality in a planar setting which is in the spirit of the Poincar\'e inequality in SBV due to De Giorgi, Carriero, Leaci. We show that for each function in SBD$^2$ one can find a modification which…
We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on…
On the hyperbolic space, we study a semilinear equation with non-autonomous nonlinearity having a critical Sobolev exponent. The Poincar\'e-Sobolev equation on the hyperbolic space explored by Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa…