Show Summary Details
More options …

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2017: 1.676

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 2.045
Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

Online
ISSN
1864-8266
See all formats and pricing
More options …
Volume 11, Issue 3

Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form

Miroslav Bulíček
• Corresponding author
• Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague, Czech Republic
• Email
• Other articles by this author:
/ Giovanni Cupini
/ Bianca Stroffolini
• Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia,80126 Napoli, Italy
• Email
• Other articles by this author:
/ Anna Verde
• Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia,80126 Napoli, Italy
• Email
• Other articles by this author:
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/acv-2016-0054

Abstract

We prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying $\left(p,q\right)$-growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q.

MSC 2010: 35J47; 35J25

References

• [1]

M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions, Lecture Notes in Math. 1818, Springer, Berlin, 2003. Google Scholar

• [2]

M. Bildhauer and M. Fuchs, ${C}^{1,\alpha }$-solutions to non-autonomous anisotropic variational problems, Calc. Var. Partial Differential Equations 24 (2005), no. 3, 309–340. Google Scholar

• [3]

F. E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 780–785.

• [4]

M. Carozza, J. Kristensen and A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 3, 395–411.

• [5]

M. Carozza, J. Kristensen and A. Passarelli di Napoli, Regularity of minimizers of autonomous convex variational integrals, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 4, 1065–1089. Google Scholar

• [6]

G. Cupini, F. Leonetti and E. Mascolo, Existence of weak solutions for elliptic systems with $p,q$-growth, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 645–658. Google Scholar

• [7]

G. Cupini, P. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under $p,q$-growth, Adv. Differential Equations 19 (2014), no. 7–8, 693–724. Google Scholar

• [8]

E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Unione Mat. Ital. (4) 1 (1968), 135–137. Google Scholar

• [9]

L. Esposito, F. Leonetti and G. Mingione, Higher integrability for minimizers of integral functionals with $\left(p,q\right)$ growth, J. Differential Equations 157 (1999), no. 2, 414–438. Google Scholar

• [10]

L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $\left(p,q\right)$ growth, Forum Math. 14 (2002), no. 2, 245–272. Google Scholar

• [11]

L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with $\left(p,q\right)$ growth, J.Differential Equations 204 (2004), no. 1, 5–55. Google Scholar

• [12]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105, Princeton University Press, Princeton, 1983. Google Scholar

• [13]

M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), no. 2, 245–248.

• [14]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, 2003. Google Scholar

• [15]

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966), 271–310.

• [16]

M. C. Hong, Some remarks on the minimizers of variational integrals with nonstandard growth conditions, Boll. Unione Mat. Ital. A (7) 6 (1992), no. 1, 91–101. Google Scholar

• [17]

F. Leonetti, Weak differentiability for solutions to nonlinear elliptic systems with $p,q$-growth conditions, Ann. Mat. Pura Appl. (4) 162 (1992), 349–366. Google Scholar

• [18]

F. Leonetti and P. V. Petricca, Regularity for solutions to some nonlinear elliptic systems, Complex Var. Elliptic Equ. 56 (2011), no. 12, 1099–1113.

• [19]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder, Bull. Soc. Math. France 93 (1965), 97–107. Google Scholar

• [20]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996. Google Scholar

• [21]

P. Marcellini, Un example de solution discontinue d’un problème variationnel dans le cas scalaire, preprint (1987).

• [22]

P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), no. 3, 267–284.

• [23]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30. Google Scholar

• [24]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), no. 4, 355–426.

• [25]

G. Mingione, Singularities of minima: A walk on the wild side of the calculus of variations, J. Global Optim. 40 (2008), no. 1–3, 209–223.

• [26]

V. Šverák and X. Yan, A singular minimizer of a smooth strongly convex functional in three dimensions, Calc. Var. Partial Differential Equations 10 (2000), no. 3, 213–221.

• [27]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150.

• [28]

S. Zhou, A note on nonlinear elliptic systems involving measures, Electron. J. Differential Equations 2000 (2000), Paper No. 08. Google Scholar

Revised: 2017-01-26

Accepted: 2017-03-15

Published Online: 2017-04-19

Published in Print: 2018-07-01

Miroslav Bulíček’s work was supported by the ERC-CZ project LL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic. Miroslav Bulíček is a member of the Nečas center for Mathematical Modeling. The other authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 3, Pages 273–288, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.