A proof of Khavinson's conjecture in $\mathbf{R}^4$

TL;DR

This paper proves Khavinson's conjecture in four-dimensional space by establishing optimal gradient estimates for bounded harmonic functions in the unit ball, solving a longstanding extremal problem.

Contribution

It provides the first proof of Khavinson's conjecture in -dimensional space, extending previous results to a higher dimension and solving a generalized extremal problem.

Findings

Established the optimal pointwise gradient estimates for harmonic functions in D

Solved the generalized Khavinson problem in D

Confirmed the conjecture for harmonic functions in the unit ball of D

Abstract

The paper deals with an extremal problem for bounded harmonic functions in the unit ball of $\mathbf{R}^4$. We solve the generalized Khavinson problem in $\mathbf{R}^4$. This precise problem was formulated by G. Kresin and V. Maz'ya for harmonic functions in the unit ball and in the half--space of $\mathbf{R}^n$. We find the optimal pointwise estimates for the norm of the gradient of bounded real--valued harmonic functions.