The notion of $\tilde{\ell }$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde{\ell }$-stable functions coincides with the class of C${}^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral ${\int }_{\Omega }F\left(\nabla u\left(x\right)\right)\mathrm{d}x$ where $\Omega \subset {ℝ}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition$F\left(\xi \right)\le c(1+|\xi {|}^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space ${ℒ}^{p,\mu }(\Omega ,{ℝ}^{N\times n})$, µ < n, Campanato space ${ℒ}^{p,n}(\Omega ,{ℝ}^{N\times n})$ and the space of bounded mean oscillation $\mathrm{BMO}\Omega ,{ℝ}^{N\times n})$. The admissible maps $u:\Omega \to {ℝ}^N$ are of Sobolev class W1,p, satisfying a Dirichlet boundary...