Related papers: Brunn-Minkowski Inequality for p-Harmonic Measures

In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, $\mbox{Cap}_{\mathcal{A}}$, where $\mathcal{A}$-capacity is associated with a nonlinear elliptic PDE whose…

We give the counter-examples related to a Gaussian Brunn-Minkowski inequality and the (B) conjecture.

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular…

We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in $\mathbb{R}^n$, we have $|A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in…

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local…

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq…

This article describes a new proof of the equality condition for the Brunn-Minkowski inequality.

In this work we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications.

We investigate various boundary decay estimates for $p(\cdot)$-harmonic functions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot)$-harmonic…

Suppose that $p \in (1,\infty]$, $\nu \in [1/2,\infty)$, $\mathcal{S}_\nu = \left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |\phi| < \frac{\pi}{2\nu}\right\}$, where $\phi$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and…

In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let $p$ be fixed with $2\leq n\leq p<\infty$.…

In this paper, we give a new generalization of the Bohr inequality in refined form both for bounded analytic functions, and for sense-preserving harmonic functions with analytic part being bounded.

We obtain two-bound estimates for the local growth of pluri-subharmonic functions in terms of Siciak and relative extremal functions. As applications, we give simple new proofs of "Bernstein doubling inequality" and the main result in…

In this paper we establish Minkowski inequality and Brunn--Minkowski inequality for $p$-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn--Minkowski inequality for quermassintegral differences of…

For positive $p$-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension $n$, $p$ and the radius of the ball on…

In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for length spaces. Our new definition based only on distance properties allows us also to deal with discrete spaces.…

For all $1<p<\infty$ and $N\ge 2$ we prove that there is a constant $\alpha(p,N)>0$ such that the $p$-harmonic measure in $\R^N_+$ of a ball of radius $0 < \delta \leq 1$ in $\R^{N-1}$ is bounded above and below by a constant times $\delta…

We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p>1. This extends previous results of Manfredi and Lindqvist for certain range of p's.

We characterize the weighted Hardy's inequalities for monotone functions in ${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our…