Nonsmooth convex functionals and feeble viscosity solutions of singular Euler–Lagrange equations

Abstract

Let $F=F(A)$ be nonnegative, convex and in $C^2(\mathbb {R}^n {\setminus }\mathcal {K})$ with $\mathcal {K}\subsetneqq \mathbb {R}^n$ a closed set. We prove that local minimisers in $(C^0\cap W^{1,1}_{\mathrm{loc}})(\Omega )$ of

$$\begin{aligned} E (u,\Omega ) := \int _{{\Omega }} F(Du), \quad \Omega \subseteq \mathbb {R}^{n}, \quad (1) \end{aligned}$$

are “very weak” viscosity solutions on $\Omega$ in the sense of Juutinen and Lindqvist [Commun PDE 30(3):305–321, 2005] of the highly singular Euler–Lagrange equation of (1) expanded:

$$\begin{aligned} F_{AA}(Du){:}\;D^{2}u = 0. \quad (2) \end{aligned}$$

The hypotheses on $F$ do not guarrantee existence of minimising weak solutions and include the singular $p$-Laplacian for $p\in (1,2)$. A much deeper converse is also true, if $\mathcal {K}=\{0\}$ and extra natural assumptions are satisfied. Our main advance is that we introduce systematic “flat” sup-convolution regularisations which apply to general singular nonlinear PDE in order to cancel the strong singularity of $F$. As an application we extend a classical theorem of calculus of variations regarding existence for the Dirichlet problem. These results extends previous work of Julin and Juutinen [Commun PDE 37(5):934–946, 2012] and Juutinen et al. (SIAM J Math Anal 33:699–717, 2001).