Abstract
We prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying
Funding statement: 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).
Acknowledgements
We thank the anonymous referee for his/her careful reading of the paper and valuable comments.
References
[1] M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions, Lecture Notes in Math. 1818, Springer, Berlin, 2003. 10.1007/b12308Search in Google Scholar
[2]
M. Bildhauer and M. Fuchs,
[3] F. E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 780–785. 10.1090/S0002-9904-1965-11391-XSearch in Google Scholar
[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. 10.1016/j.anihpc.2011.02.005Search in Google Scholar
[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. 10.2422/2036-2145.201208_005Search in Google Scholar
[6]
G. Cupini, F. Leonetti and E. Mascolo,
Existence of weak solutions for elliptic systems with
[7]
G. Cupini, P. Marcellini and E. Mascolo,
Existence and regularity for elliptic equations under
[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. Search in Google Scholar
[9]
L. Esposito, F. Leonetti and G. Mingione,
Higher integrability for minimizers of integral functionals with
[10]
L. Esposito, F. Leonetti and G. Mingione,
Regularity results for minimizers of irregular integrals with
[11]
L. Esposito, F. Leonetti and G. Mingione,
Sharp regularity for functionals with
[12] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105, Princeton University Press, Princeton, 1983. 10.1515/9781400881628Search in Google Scholar
[13] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), no. 2, 245–248. 10.1007/BF01158049Search in Google Scholar
[14] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, 2003. 10.1142/5002Search in Google Scholar
[15] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966), 271–310. 10.1007/BF02392210Search in Google Scholar
[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. Search in Google Scholar
[17]
F. Leonetti,
Weak differentiability for solutions to nonlinear elliptic systems with
[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. 10.1080/17476933.2010.487208Search in Google Scholar
[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. 10.24033/bsmf.1617Search in Google Scholar
[20] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996. 10.1007/978-1-4899-6824-1Search in Google Scholar
[21] P. Marcellini, Un example de solution discontinue d’un problème variationnel dans le cas scalaire, preprint (1987). Search in Google Scholar
[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. 10.1007/BF00251503Search in Google Scholar
[23]
P. Marcellini,
Regularity and existence of solutions of elliptic equations with
[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. 10.1007/s10778-006-0110-3Search in Google Scholar
[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. 10.1007/s10898-007-9226-1Search in Google Scholar
[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. 10.1007/s005260050151Search in Google Scholar
[27] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. 10.1016/0022-0396(84)90105-0Search in Google Scholar
[28] S. Zhou, A note on nonlinear elliptic systems involving measures, Electron. J. Differential Equations 2000 (2000), Paper No. 08. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Local regularity and compactness for the p-harmonic map heat flows
- 𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN
- Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form
- On a new class of fractional partial differential equations II
- On the structure of flat chains modulo p
- A remark on the two-phase obstacle-type problem for the p-Laplacian
Articles in the same Issue
- Frontmatter
- Local regularity and compactness for the p-harmonic map heat flows
- 𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN
- Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form
- On a new class of fractional partial differential equations II
- On the structure of flat chains modulo p
- A remark on the two-phase obstacle-type problem for the p-Laplacian