Strict convexity and the regularity of solutions to variational problems

Cellina, Arrigo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 862-871.

We consider the problem of minimizing

$${\int }_{\Omega }[L\left(\nabla v\left(x\right)\right)+g(x,v\left(x\right))]dx\text{on}{u}^0+W_0^{1,2}\left(\Omega \right)$$

where $\Omega$ is a bounded open subset of ${ℝ}^N$ and $L$ is a convex function that grows quadratically outside the unit ball, while, when $\left|\nabla v\right|<1$, it behaves like ${\left|\nabla v\right|}^p$ with $1<p<2$. We show that, for each $\omega \subset \subset \Omega$, there exists a constant $H$, depending on $\omega$ but not on $p$, such that both

$${\parallel \nabla u\parallel }_{W^{1,2}\left(\omega \right)}\le H\text{and}{\parallel \frac{\nabla u}{{\left|\nabla u\right|}^{2-p}}\parallel }_{W^{1,2}\left(\omega \right)}\le \frac{H}{{(p-1)}^2};$$

in particular, for every $i=1,...N$, we have $max\{\frac{|u_{x_i}|}{{\left|\nabla u\right|}^{2-p}},|u_{x_i}\left|\right\}\in W_{loc}^{1,2}\left(\Omega \right)$.

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