# A proof of the Khavinson conjecture

## TL;DR

This paper proves the Khavinson conjecture in three dimensions, establishing a sharp inequality for the gradient of bounded harmonic functions in the unit ball, resolving an open problem posed in 1992.

## Contribution

It provides the first proof of the Khavinson conjecture in \\mathbb{R}^3, offering a precise gradient bound that improves previous partial results.

## Key findings

- Established a sharp gradient inequality for harmonic functions in 3D
- Resolved the open Khavinson conjecture in three dimensions
- Enhanced understanding of extremal problems for harmonic functions

## Abstract

\begin{abstract} This paper deals with an extremal problem for bounded harmonic functions in the unit ball $\mathbb{B}^n.$ We solve the Khavinson conjecture in $\mathbb{R}^3,$ an intriguing open question since 1992 posed by D. Khavinson, later considered in a general context by Kresin and Maz'ya. Precisely, we obtain the following inequality: $$|\nabla u(x)|\leq \frac{1}{\rho^2}\bigg({\frac{(1+\frac{1}{3}\rho^2)^{\frac{3}{2}}}{1-\rho^2}-1}\bigg) \sup_{|y|<1} |u(y)|, $$ with $\rho=|x|,$ thus sharpening the previously known with $|\langle \nabla u(x),n_{x} \rangle |$ instead of $|\nabla u(x)|, $ where $n_{x}=\frac{x}{|x|}.$ \end{abstract}