Sobolev’s original definition of his spaces revisited and a comparison with nowadays definition. (English) Zbl 0947.46022

The theme of this interesting report is the comparison between Sobolev’s definition \[ L^{m,p}(\Omega)= \{u\in L^1_{\text{loc}}(\Omega): \partial^\alpha u\in L^p(\Omega),|\alpha|= m\} \] and the definition used nowadays \[ W^{m,p}(\Omega)= \{u\in L^p(\Omega): \partial^\alpha u\in L^p(\Omega),|\alpha|\leq m\} \] of the so-called Sobolev space. The author recalls both definitions and illustrates with examples the problems that occur with the presently used definition, for instance when one considers the Dirichlet problem in an unbounded domain or when one introduces a cancellation condition in the definition of the space.
The author also lists several problems that arise in the definition of \(L^{m,p}(\Omega)\), concerning existence of intermediate derivatives, completeness, and equivalence of norms. Part of the paper is dedicated to present recent joint work with J. Naumann, published in Math. Bohem. 124, No. 2-3, 315-328 (1999). This joint work present a method to overcome the difficulties in Sobolev’s definition mentioned above. The main ingredient is to prove a Poincaré inequality for balls or cubes.
The last part of the paper under review proves that \(L^{1,p}(\Omega)= W^{1,p}(\Omega)\), assuming weak conditions on the domain. Finally, the author proves an example of a bounded domain \(\Omega\subset\mathbb{R}^2\) for which \(W^{1,p}(\Omega)\) is strictly contained in \(L^{1,p}(\Omega)\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems