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:
[1]

B. AvelinT. 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

[2]

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

[3]

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

[4]

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

[5]

E. Di NezzaG. 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

[6]

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

[7]

E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994.  Google Scholar

[8]

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

[9]

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

[10]

W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

B. AvelinT. 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

[2]

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

[3]

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

[4]

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

[5]

E. Di NezzaG. 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

[6]

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

[7]

E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994.  Google Scholar

[8]

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

[9]

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

[10]

W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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