American Institute of Mathematical Sciences

Advanced
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

Arrigo Cellina

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 & 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.  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 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

[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. 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

[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 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

[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

[1]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
[2]

Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371
[3]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[4]

John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533
[5]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[6]

Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020
[7]

Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
[8]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021, 11 (3) : 658-679. doi: 10.3934/mcrf.2021017
[9]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[10]

Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
[11]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
[12]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922
[13]

Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[14]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[15]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083
[16]

Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063
[17]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[18]

Carlo Mercuri, Michel Willem. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 469-493. doi: 10.3934/dcds.2010.28.469
[19]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[20]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (141)
  • HTML views (142)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences