Singular set estimates in quasilinear Alt-Caffarelli problems

For a nonnegative, uniformly convex function $Φ$, we study nonnegative minimizers of the quasilinear Alt-Caffarelli energy $\int_ΩΦ(\nabla u)+χ_{\{u>0\}}$. Under mild regularity assumptions on $Φ$, we show that the codimension of the singular set of the free boundary is at least $2+η$ for a positive $η$. With a singular minimizer in three dimensions in a forthcoming work, this result is sharp as long as integer dimensions are considered. Our proof method is flexible, and we use it to also prove a similar result for minimizers of the $p$-Laplacian counterpart of the Alt-Caffarelli energy.