# The Leibniz rule for the Dirichlet and the Neumann Laplacian

## TL;DR

This paper investigates bilinear estimates in Sobolev spaces for Dirichlet and Neumann Laplacians, revealing optimal regularity conditions in half spaces and emphasizing boundary behavior's role in these estimates.

## Contribution

It establishes the optimal regularity for bilinear estimates involving Dirichlet and Neumann Laplacians in half spaces, focusing on boundary value handling.

## Key findings

- Optimal regularity conditions identified for bilinear estimates
- Boundary behavior significantly influences estimate accuracy
- Method developed for boundary value management in Sobolev spaces

## Abstract

We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.