相关论文: On monotonicity of some functionals under rearrang…
We give necessary and sufficient conditions for the P\'olya--Szeg\"o type inequality with variable exponent of summability.
We analyze the Steiner rearrangement in any codimension of Sobolev and $BV$ functions. In particular, we prove a P\'olya-Szeg\H{o} inequality for a large class of convex integrals. Then, we give minimal assumptions under which functions…
The Polya-Szeg\H{o} inequality in $\mathbb{R}^n$ states that, given a non-negative function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$, its spherically symmetric decreasing rearrangement $f^*:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$ is…
In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function $u^*$ with a function $u$ defined on the integers and prove the corresponding…
This paper is devoted to a kind of rearrangement of functions on CD(k,n)-spaces, which satisfy a Polya-Szeg\"o type inequality. We use this rearrangement to prove the validity of a Moser-Trudinger type inequality on a wide class of metric…
The transformations of functions acting on sublevel sets that satisfy a P\'olya-Szeg\H{o} inequality are characterized as those being induced by transformations of sets that do not increase the associated capacity.
Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz…
We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional…
An affine rearrangement inequality is established which strengthens and implies the recently obtained affine P\'olya--Szeg\"o symmetrization principle for functions on $\mathbb{R}^n$. Several applications of this new inequality are derived.…
A P\'olya-Szeg\"o inequality for the circular rearrangement is proven, under general assumptions. In addition, sufficient conditions are given, under which all the extremals of the inequality are symmetric.
The symmetric decreasing rearrangement of functions on $\mathbb{R}^n$ features in several seminal inequalities, such as the P\'olya-Szeg\H{o} inequality. The latter was shown by the authors to hold for all smoothing rearrangements, a class…
The goal of this note is to fill a gap in the proof of the first two items of Theorem 5.1 in [4], which relies on Polya type inequalities and the characterization of the equality cases for monotone rearrangements given in Propositions 4.1…
In this thesis, we study problems at the interface of analysis and discrete mathematics. We discuss analogues of well known Hardy-type inequalities and Rearrangement inequalities on the lattice graphs $\mathbb{Z}^d$, with a particular focus…
This paper deals with the behavior of the periodic Gagliardo seminorm under two types of rearrangements, namely under a periodic, and respectively a cylindrical, symmetric decreasing rearrangement. Our two main results are P\'olya-Szeg\H{o}…
We prove that a nonlocal functional approximating the standard Dirichlet $p$-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness, and the…
We prove some P\'olya-Szeg\"o type inequalities which involve couples of functions and their rearrangements. Our inequalities reduce to the classical P\'olya-Szeg\"o principle when the two functions coincide. As an application, we give a…
The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands…
We prove a weak-strong convergence result for functionals of the form $\int_{\mathbb{R}^N} j(x, u, Du)\,dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya-Szeg\"o…
This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and P\'{o}lya-Szeg\H{o} inequalities. Then we…
For any $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ the symmetric decreasing rearrangement $f^*$ satisfies the Polya-Szeg\H{o} inequality $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$. The goal of this paper is to establish…