    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
##### References:
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show all references

##### References:
  B. Avelin, T. Kuusi and G. Mingione, Nonlinear Calderon-Zygmund theory in the limiting case, Arch. Rat. Mech. Anal., 227 (2018), 663-714. doi: 10.1007/s00205-017-1171-7.  Google Scholar  A. Cellina, The regularity of solutions to some variational problems, including the p-Laplace equation for 2 ≤ p < 3, ESAIM: COCV, 23 (2017), 1543-1553. doi: 10.1051/cocv/2016064.  Google Scholar  A. Cianchi and V. G. Maz'ya, Second-order two-sided estimates in nonlinear elliptic problems, Archive for Rational Mechanics and Analysis, (2017), 1-31. doi: 10.1007/s00205-018-1223-7. Google Scholar  F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, Heidelberg, 2012. doi: 10.1007/978-1-4471-2807-6.  Google Scholar  E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar  L. Esposito and G. Mingione, Some remarks on the regularity of weak solutions of degenerate elliptic systems, Rev. Mat. Complutense, 11 (1998), 203-219. Google Scholar  E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994. Google Scholar  O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian. Academic Press, New York-London, 1968. Google Scholar  J. J. Manfredi and A. Weitsman, On the Fatou Theorem for p-harmonic functions, Comm. Partial Differential Equations, 13 (1988), 651-668. doi: 10.1080/03605308808820556.  Google Scholar  W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar
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