What is it about?
This article studies when every function in a generalized Sobolev space, called an Orlicz–Sobolev space, can be approximated by smooth functions whose derivatives stay uniformly bounded. The authors prove that on any bounded simply connected planar domain, and for any doubling Young function, smooth functions with bounded derivatives are dense in the corresponding Orlicz–Sobolev space when the norm only controls the highest-order derivatives.
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Why is it important?
Density of smooth functions is a fundamental issue in the theory of function spaces because it justifies many standard tools, such as approximation by smooth test functions and the use of variational methods. For Orlicz–Sobolev spaces, which generalize the classical Sobolev spaces to allow more flexible growth conditions via Young functions, such density results are much less understood, especially beyond first order. The result clarifies how the geometry of planar domains interacts with the structure of Orlicz–Sobolev spaces and shows that, under the doubling assumption on the Young function, one still has good approximation by smooth, bounded-derivative functions when the norm only involves the top-order derivatives
Perspectives
This work grew out of my interest in understanding how far classical Sobolev approximation techniques can be pushed in more general function spaces that naturally appear in nonlinear analysis and mappings of finite distortion. I have been particularly motivated by the fact that, although smooth functions are known to be dense in many Sobolev and Orlicz–Sobolev spaces, the behavior of derivatives near the boundary can obstruct the density of nicer subspaces, and this article identifies a setting where one can still obtain strong density of smooth functions with bounded derivatives. The project also led me to several open questions, especially for higher-order Sobolev and Orlicz–Sobolev spaces with full norms, which continue to guide my ongoing work in the analysis of function spaces on irregular domains.
Walter Andrés Ortíz Vargas
Universidad Internacional de La Rioja
Read the Original
This page is a summary of: A density result on Orlicz-Sobolev spaces in the plane, Journal of Mathematical Analysis and Applications, November 2021, Elsevier,
DOI: 10.1016/j.jmaa.2021.125329.
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