Non-existence of a Wente's $L^\infty$ estimate for the Neumann problem

TL;DR

This paper demonstrates that Wente's inequality does not hold for Neumann and Robin boundary conditions through counterexamples, highlighting limitations of the estimate in these contexts.

Contribution

The paper provides the first known counterexamples showing the failure of Wente's $L^ abla$ estimate for Neumann and Robin boundary conditions.

Findings

Wente's inequality fails for Neumann boundary conditions.

Wente's estimates do not hold for Robin boundary conditions.

Counterexamples illustrate the limitations of Wente's estimate.

Abstract

We provide a counterexample of Wente's inequality in the context of Neumann boundary conditions. We will also show that Wente's estimates fails for general boundary conditions of Robin type.