The Laplacian on Cartesian products with mixed boundary conditions

TL;DR

This paper defines the Laplacian on Cartesian product domains with mixed boundary conditions using quadratic forms, establishing its consistency with standard definitions and deriving Sobolev space tensor representations.

Contribution

It introduces a quadratic form-based definition of the Laplacian for mixed boundary conditions and proves its consistency with classical definitions.

Findings

Tensor representations of Sobolev spaces are derived.

A criterion for second-order Sobolev domain membership is established.

The definition aligns with standard Laplacian for homogeneous boundary conditions.

Abstract

A definition of the Laplacian on Cartesian products with mixed boundary conditions using quadratic forms is proposed. Its consistency with the standard definition for homogeneous and certain mixed boundary conditions is proved and, as a consequence, tensor representations of the corresponding Sobolev spaces of first order are derived. Moreover, a criterion for the domain to belong to the Sobolev space of second order is proved.