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Non-Lipschitz minimizers of smooth uniformly convex functionals
Vladimír Sverák1, Xiaodong Yan
1Department of Mathematics, University of Minnesota, Minneapolis 55455, USA. sverak@math.umn.edu
Summary
This study introduces a novel method for constructing non-Lipschitz minimizers for specific mathematical functionals. The technique utilizes null Lagrangians to achieve these complex solutions in calculus of variations.
Area of Science:
- Mathematical analysis
- Calculus of variations
- Optimization theory
Background:
- Smooth, uniformly convex functionals are crucial in various mathematical and physical models.
- Finding non-Lipschitz minimizers for these functionals presents significant theoretical challenges.
- Existing methods often struggle with the regularity of solutions.
Purpose of the Study:
- To develop a method for constructing non-Lipschitz minimizers.
- To address the challenge of finding solutions with limited regularity for convex functionals.
- To extend the applicability of calculus of variations techniques.
Main Methods:
- Construction of non-Lipschitz minimizers.
- Application of null Lagrangians.
- Analysis of integral functionals of the form I(u) = integral (Omega) f(Du(x))dx.
Main Results:
- Successfully constructed non-Lipschitz minimizers for the specified class of functionals.
- Demonstrated the efficacy of null Lagrangians in this construction.
- Provided a new approach to solving problems in the calculus of variations.
Conclusions:
- The use of null Lagrangians provides an effective tool for constructing non-Lipschitz solutions.
- This method expands the understanding of minimizer regularity for convex functionals.
- Opens new avenues for research in optimization and partial differential equations.
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