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

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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.

Keywords: Second order derivatives, p-laplacian.

Mathematics Subject Classification: 49K10, 35J25, 35J60, 35B65.

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

References:

[1]	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.
[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.
[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.
[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.
[5]	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.
[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.
[7]	E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994.
[8]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian. Academic Press, New York-London, 1968.
[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.
[10]	W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:

[1]	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.
[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.
[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.
[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.
[5]	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.
[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.
[7]	E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994.
[8]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian. Academic Press, New York-London, 1968.
[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.
[10]	W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.

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