Validated bounds on embedding constants for Sobolev space Banach algebras

Abstract

Sobolev spaces and their embedding properties have long been of central importance in the study of partial differential equations, particularly for the classical theoretical analysis of both linear and nonlinear problems. In recent years, computer‐assisted proof techniques have been developed for obtaining existence and uniqueness proofs of solutions to a variety of nonlinear partial differential equations that arise in applications, and they have led to a number of new results that currently lie beyond the reach of classical analytical approaches. The use of computer‐assisted methods, however, frequently relies on the explicit knowledge of a variety of embedding constants for Sobolev spaces. In the present paper, we show that in the context of certain Sobolev space Banach algebras, these constants themselves can be bounded rigorously and precisely using validated computations based on interval arithmetic, combined with analytical error estimates.