The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4

We consider the higher differentiability of solutions to the problem of minimising

$\int_{Ω} [L(\nabla v(x))+g(x, v(x))]dx~~~ \hbox {on}~~~ u^0+W^{1, p}_0(Ω)$

where $\Omega\subset \mathbb R^N$, $L(ξ) = l(|ξ|) = \frac{1}{p}|ξ|^p$ and $ u^0∈ W^{1, p}(Ω)$ and hence, in particular, the higher differentiability of weak solution to the equation

${\rm div }(|\nabla u|^ {p-2}\nabla u) = f.$

We show that, for $3≤ p < 4$, under suitable assumptions on $g$, there exists a solution $ u^*$ to the Euler-Lagrange equation associated to the minimisation problem, such that

$\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$

for $0 < s < 4-p$. In particular, for $p = 3$, we show that the solution $u^*$ is such that $\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$ for every $s < 1$. This result is independent of $N$. We present an example for $N = 1$ and $p = 3$ whose solution $u$ is such that $\nabla u^*$ is not in $W^{1, 2}_{loc}(\Omega)$, thus showing that our result is sharp.