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A surprising formula for Sobolev norms
Haïm Brezis1,2,3,4, Jean Van Schaftingen5, Po-Lam Yung6,7
1Department of Mathematics, Rutgers University, Piscataway, NJ 08854; brezis@math.rutgers.edu.
This study demonstrates a key equivalence in Sobolev and Gagliardo seminorms, replacing strong derivatives with weak ones. This finding offers new estimation methods for fractional inequalities in specific mathematical cases.
Area of Science:
- Mathematical Analysis
- Functional Analysis
- Partial Differential Equations
Background:
- The Gagliardo-Nirenberg and fractional Sobolev inequalities are fundamental in analysis.
- Estimating these inequalities often relies on strong derivatives.
- Challenges arise in cases requiring weak derivative formulations.
Purpose of the Study:
- To establish a novel equivalence between Sobolev and Gagliardo seminorms.
- To explore the implications of using weak derivatives in these seminorms.
- To develop alternative estimation techniques for specific mathematical scenarios.
Main Methods:
- Mathematical equivalence proof.
- Analysis of Sobolev and Gagliardo seminorms.
- Application of weak derivative concepts.
Main Results:
- Equivalence established between Sobolev seminorm [Formula: see text] and a modified Gagliardo seminorm using weak derivatives.
- Derivation of alternative estimates in exceptional cases.
- Demonstration of applicability where standard inequalities fail.
Conclusions:
- The established equivalence provides a new perspective on Sobolev and Gagliardo seminorms.
- The derived estimates offer valuable tools for analysis in challenging mathematical contexts.
- This work contributes to a deeper understanding of fractional inequalities and their behavior.
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