Solution to the Khavinson problem near the boundary of the unit ball

TL;DR

This paper establishes sharp boundary estimates for the gradient of bounded harmonic functions in the unit ball, confirming a conjecture by Khavinson about the equality of certain sharp constants near the boundary.

Contribution

It provides the first proof that the sharp constants for the radial derivative and the gradient modulus coincide near the boundary of the unit ball, partially confirming Khavinson's conjecture.

Findings

Sharp boundary estimates for harmonic function gradients

Equality of sharp constants near the boundary

Partial confirmation of Khavinson's conjecture

Abstract

This paper deals with an extremal problem for harmonic functions in the unit ball of $\mathbf{R}^n$. We are concerned with the pointwise sharp estimates for the gradient of real--valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson.