WILD SOLUTIONS TO SCALAR EULER-LAGRANGE EQUATIONS

. We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W1,1 solutions are necessarily W 1,2 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 W1,1 solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W1,1 solution to be improved to W 1,2 suitable assumptions on the functional and solution.