No Lavrentiev gap for some double phase integrals. (English) Zbl 1530.49023
Summary: We prove the absence of the Lavrentiev gap for non-autonomous functionals
\[
\mathcal{F}(u) := \int\limits_{\Omega}f(x,Du(x))\,dx,
\]
where the density \(f(x,z)\) is \(\alpha \)-Hölder continuous with respect to \(x \in \Omega \subset\mathbb{R}^n \), it satisfies the \((p,q)\)-growth conditions
\[
\vert z\vert^p \leqslant f(x,z) \leqslant L(1+\vert z\vert^q),
\]
where \(1<p<q<p(\frac{n+\alpha}{n})\), and it can be approximated from below by suitable densities \(f_k \).
MSC:
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35J47 | Second-order elliptic systems |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35J20 | Variational methods for second-order elliptic equations |
References:
| [1] | E. Acerbi and N. Fusco, Regularity of minimizers of non-quadratic functionals: The case \(1<p<2\), J. Math. Anal. Appl. 140 (1989), 115-135. · Zbl 0686.49004 |
| [2] | R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. · Zbl 0314.46030 |
| [3] | G. Alberti and P. Majer, Gap phenomenon for some autonomous functionals, J. Convex Anal. 1 (1994), no. 1, 31-45. · Zbl 0817.49013 |
| [4] | A. K. Balci, L. Diening and M. Surnachev, New examples on Lavrentiev gap using fractals, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 180. · Zbl 1453.35082 |
| [5] | P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 62. · Zbl 1394.49034 |
| [6] | G. Buttazzo, The gap phenomenon for integral functionals: Results and open questions, Variational Methods, Nonlinear Analysis and Differential Equations: Proceedings of the International workshop for the 75-th birthday of Cecconi, Edizioni Culturali Internazionali, Genova (1993), 50-59. |
| [7] | G. Buttazzo and M. Belloni, A survey on old and recent results about the gap phenomenon in the calculus of variations, Recent Developments in Well-Posed Variational Problems, Math. Appl. 331, Kluwer Academic, Dordrecht (1995), 1-27. · Zbl 0852.49001 |
| [8] | G. Buttazzo and V. J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal. 110 (1992), no. 2, 434-460. · Zbl 0784.49006 |
| [9] | P. Celada, G. Cupini and M. Guidorzi, Existence and regularity of minimizers of nonconvex integrals with \(p-q\) growth, ESAIM Control Optim. Calc. Var. 13 (2007), no. 2, 343-358. · Zbl 1124.49031 |
| [10] | I. Chlebicka, P. Gwiazda and A. Zatorska-Goldstein, Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, 1431-1465. · Zbl 1419.35092 |
| [11] | I. Chlebicka, P. Gwiazda and A. Zatorska-Goldstein, Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon, J. Differential Equations 267 (2019), no. 2, 1129-1166. · Zbl 1430.35123 |
| [12] | M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219-273. · Zbl 1325.49042 |
| [13] | M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443-496. · Zbl 1322.49065 |
| [14] | G. Cupini, F. Giannetti, R. Giova and A. Passarelli di Napoli, Regularity results for vectorial minimizers of a class of degenerate convex integrals, J. Differential Equations 265 (2018), no. 9, 4375-4416. · Zbl 1411.49021 |
| [15] | G. Cupini, M. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with p-q growth, Nonlinear Anal. 54 (2003), no. 4, 591-616. · Zbl 1027.49032 |
| [16] | R. De Arcangelis, Some remarks on the identity between a variational integral and its relaxed functional, Ann. Univ. Ferrara Sez. VII (N. S.) 35 (1989), 135-145. · Zbl 0715.49002 |
| [17] | C. De Filippis and G. Mingione, A borderline case of Calderón-Zygmund estimates for nonuniformly elliptic problems, Algebra i Analiz 31 (2019), no. 3, 82-115. · Zbl 1435.35127 |
| [18] | C. De Filippis and G. Mingione, On the regularity of minima of non-autonomous functionals, J. Geom. Anal. 30 (2020), no. 2, 1584-1626. · Zbl 1437.35292 |
| [19] | C. De Filippis and G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Ration. Mech. Anal. 242 (2021), no. 2, 973-1057. · Zbl 1483.49050 |
| [20] | M. Eleuteri, P. Marcellini and E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl. (4) 195 (2016), no. 5, 1575-1603. · Zbl 1354.35035 |
| [21] | M. Eleuteri, P. Marcellini and E. Mascolo, Regularity for scalar integrals without structure conditions, Adv. Calc. Var. 13 (2020), no. 3, 279-300. · Zbl 1445.35159 |
| [22] | A. Esposito, F. Leonetti and P. Vincenzo Petricca, Absence of Lavrentiev gap for non-autonomous functionals with \((p,q)\)-growth, Adv. Nonlinear Anal. 8 (2019), no. 1, 73-78. · Zbl 1417.49050 |
| [23] | L. Esposito, F. Leonetti and G. Mingione, Higher integrability for minimizers of integral functionals with \((p,q)\) growth, J. Differential Equations 157 (1999), no. 2, 414-438. · Zbl 0939.49021 |
| [24] | L. Esposito, F. Leonetti and G. Mingione, Regularity for minimizers of functionals with p-q growth, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 2, 133-148. · Zbl 0928.35044 |
| [25] | L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with \((p,q)\) growth, J. Differential Equations 204 (2004), no. 1, 5-55. · Zbl 1072.49024 |
| [26] | I. Fonseca, J. Malý and G. Mingione, Scalar minimizers with fractal singular sets, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 295-307. · Zbl 1049.49015 |
| [27] | M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 3, 185-208. · Zbl 0594.49004 |
| [28] | E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. · Zbl 1028.49001 |
| [29] | C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. 431 (1992), 7-64. · Zbl 0776.35006 |
| [30] | L. Koch, Global higher integrability for minimisers of convex functionals with \((p,q)\)-growth, Calc. Var. Partial Differential Equations 60 (2021), no. 2, Paper No. 63. · Zbl 1461.49017 |
| [31] | T. Kuusi and G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 4, 929-1004. · Zbl 1394.35206 |
| [32] | M. Lavrentieff, Sur quelques problèmes du calcul des variations, Ann. Mat. Pura Appl. (4) 4 (1927), no. 1, 7-28. · JFM 53.0481.02 |
| [33] | J. J. Manfredi, Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations, Ph.D. thesis, University of Washington, St. Louis, 1986. |
| [34] | J. J. Manfredi, Regularity for minima of functionals with p-growth, J. Differential Equations 76 (1988), no. 2, 203-212. · Zbl 0674.35008 |
| [35] | P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 391-409. · Zbl 0609.49009 |
| [36] | P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal. 105 (1989), no. 3, 267-284. · Zbl 0667.49032 |
| [37] | P. Marcellini, Regularity and existence of solutions of elliptic equations with \(p,q\)-growth conditions, J. Differential Equations 90 (1991), no. 1, 1-30. · Zbl 0724.35043 |
| [38] | P. Marcellini, Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, J. Math. Anal. Appl. 501 (2021), no. 1, Paper No. 124408. · Zbl 1512.35301 |
| [39] | G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), no. 4, 355-426. · Zbl 1164.49324 |
| [40] | G. Mingione and V. Rǎdulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), no. 1, Paper No. 125197. · Zbl 1467.49003 |
| [41] | J. Ok, Regularity of ω-minimizers for a class of functionals with non-standard growth, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 48. · Zbl 1369.49050 |
| [42] | V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3 (1995), no. 2, 249-269. · Zbl 0910.49020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
