  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.
