# American Institute of Mathematical Sciences

August  2018, 38(8): 4071-4085. doi: 10.3934/dcds.2018177

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

 Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, I-20125 Milano, Italy

Received  November 2017 Revised  March 2018 Published  May 2018

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.
Citation: Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177
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