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Wild solutions to scalar Euler-Lagrange equations
- Publication Year :
- 2023
-
Abstract
- We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether $W^{1,1}$ solutions are necessarily $W^{1,2}_{\operatorname{loc}}$, which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl's classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist $W^{1,1}$ solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a $W^{1,1}$ solution to be improved to $W^{1,2}_{\operatorname{loc}}$ under suitable assumptions on the functional and solution.<br />Comment: 24 pages, 1 figure
- Subjects :
- Mathematics - Analysis of PDEs
35D30, 35A09, 35J60, 35A02
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2303.07298
- Document Type :
- Working Paper