Optimality of embeddings in Orlicz spaces

Working with function spaces in various branches of mathematical analysis introduces optimality problems, where the question of choosing a function space both accessible and expressive becomes a nontrivial exercise. A good middle ground is provided by Orlicz spaces, parameterized by a single Young function and thus accessible, yet expansive. In this work, we study optimality problems on Sobolev embeddings in Mazya classes of Euclidean domains which are defined through their isoperimetric behavior. In particular, we prove the nonexistence of optimal Orlicz spaces in certain Orlicz Sobolev embeddings in a limiting, or critical, state whose pivotal special case is the celebrated embedding of Brezis and Wainger for John domains.