A limiting case in partial regularity for quasiconvex functionals

Mirco Piccinini; Mirco Piccinini

doi:10.3934/mine.2024001

Mathematics in Engineering

2024, Volume 6, Issue 1: 1-27. doi: 10.3934/mine.2024001

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A limiting case in partial regularity for quasiconvex functionals

Mirco Piccinini ^,

Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Campus, Parco Area delle Scienze, 53/a, 43124 Parma, Italy

Received: 10 August 2023 Revised: 24 December 2023 Accepted: 24 December 2023 Published: 28 December 2023

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $ p $-growth of the type

$ w\mapsto \int \left[F(Dw)-f\cdot w\right]{\,{{\rm{d}}}x} $

feature almost everywhere $ \mbox{BMO} $-regular gradient provided that $ f $ belongs to the borderline Marcinkiewicz space $ L(n, \infty) $.
- regularity,
- quasiconvex functionals,
- degenerate variational integrals
Citation: Mirco Piccinini. A limiting case in partial regularity for quasiconvex functionals[J]. Mathematics in Engineering, 2024, 6(1): 1-27. doi: 10.3934/mine.2024001

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Abstract

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $ p $-growth of the type

$ w\mapsto \int \left[F(Dw)-f\cdot w\right]{\,{{\rm{d}}}x} $

feature almost everywhere $ \mbox{BMO} $-regular gradient provided that $ f $ belongs to the borderline Marcinkiewicz space $ L(n, \infty) $.

References

[1]	E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125–145. https://doi.org/10.1007/BF00275731 doi: 10.1007/BF00275731
[2]	E. Acerbi, N. Fusco, A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal., 99 (1987), 261–281. https://doi.org/10.1007/BF00284509 doi: 10.1007/BF00284509
[3]	E. Acerbi, G. Mingione, Regularity results for a class of quasiconvex functionals with nonstandard growth, Ann. Sc. Norm. Super. Pisa Cl. Sci., 30 (2001), 311–339.
[4]	J. M. Ball, F. Murat, $W^{1, p}$-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (1984), 225–253. https://doi.org/10.1016/0022-1236(84)90041-7 doi: 10.1016/0022-1236(84)90041-7
[5]	M. Bärlin, F. Gmeineder, C. Irving, J. Kristensen, $\mathcal{A}$-harmonic approximation and partial regularity, revisited, arXiv, 2022. https://doi.org/10.48550/arXiv.2212.12821
[6]	P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differ. Equ., 53 (2015), 803–846. https://doi.org/10.1007/s00526-014-0768-z doi: 10.1007/s00526-014-0768-z
[7]	S. S. Byun, Y. Youn, Potential estimates for elliptic systems with subquadratic growth, J. Math. Pures Appl., 131 (2019), 193–224. https://doi.org/10.1016/j.matpur.2019.02.012 doi: 10.1016/j.matpur.2019.02.012
[8]	M. Carozza, N. Fusco, G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura Appl., 175 (1998), 141–164. https://doi.org/10.1007/BF01783679 doi: 10.1007/BF01783679
[9]	A. Cianchi, Maximizing the $L^\infty$-norm of the gradient of solutions to the Poisson equation, J. Geom. Anal., 2 (1992), 499–515. https://doi.org/10.1007/BF02921575 doi: 10.1007/BF02921575
[10]	A. Cianchi, V. G. Maz'ya, Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Rational Mech. Anal., 212 (2014), 129–177. https://doi.org/10.1007/s00205-013-0705-x doi: 10.1007/s00205-013-0705-x
[11]	A. Cianchi, V. G. Maz'ya, Optimal second-order regularity for the $p$-Laplace system, J. Math. Pures Appl., 132 (2019), 41–78. https://doi.org/10.1016/j.matpur.2019.02.015 doi: 10.1016/j.matpur.2019.02.015
[12]	C. De Filippis, Quasiconvexity and partial regularity via nonlinear potentials, J. Math. Pures Appl., 163 (2022), 11–82. https://doi.org/10.1016/j.matpur.2022.05.001 doi: 10.1016/j.matpur.2022.05.001
[13]	C. De Filippis, M. Piccinini, Borderline global regularity for nonuniformly elliptic systems, Int. Math. Res. Not., 2023 (2023), 17324–17376. https://doi.org/10.1093/imrn/rnac283 doi: 10.1093/imrn/rnac283
[14]	C. De Filippis, B. Stroffolini, Singular multiple integrals and nonlinear potentials, J. Funct. Anal., 285 (2023), 109952. https://doi.org/10.1016/j.jfa.2023.109952 doi: 10.1016/j.jfa.2023.109952
[15]	L. Diening, F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 20 (2008), 523–556. https://doi.org/10.1515/FORUM.2008.027 doi: 10.1515/FORUM.2008.027
[16]	L. Diening, D. Lengeler, B. Stroffolini, A. Verde, Partial regularity for minimizers of quasi-convex functionals with general growth, SIAM J. Math. Anal., 44 (2012), 3594–3616. https://doi.org/10.1137/120870554 doi: 10.1137/120870554
[17]	L. Diening, B. Stroffolini, A. Verde, The $ \varphi$-harmonic approximation and the regularity of $ \varphi$-harmonic maps, J. Differ. Equations, 253 (2012), 1943–1958. https://doi.org/10.1016/j.jde.2012.06.010 doi: 10.1016/j.jde.2012.06.010
[18]	H. Dong, H. Zhu, Gradient estimates for singular $p$-Laplace type equations with measure data, J. Eur. Math. Soc., 2023, 1–47. https://doi.org/10.4171/jems/1400 doi: 10.4171/jems/1400
[19]	F. Duzaar, G. Mingione, Regularity for degenerate elliptic problems via $p$-harmonic approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 735–766. https://doi.org/10.1016/j.anihpc.2003.09.003 doi: 10.1016/j.anihpc.2003.09.003
[20]	F. Duzaar, G. Mingione, The $p$-harmonic approximation and the regularity of $p$-harmonic maps, Calc. Var. Partial Differ. Equ., 20 (2004), 235–256. https://doi.org/10.1007/s00526-003-0233-x doi: 10.1007/s00526-003-0233-x
[21]	F. Duzaar, K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math., 546 (2002), 73–138. https://doi.org/10.1515/crll.2002.046 doi: 10.1515/crll.2002.046
[22]	L. C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal., 95 (1986), 227–252. https://doi.org/10.1007/BF00251360 doi: 10.1007/BF00251360
[23]	M. Giaquinta, G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55–99. https://doi.org/10.1007/BF01172492 doi: 10.1007/BF01172492
[24]	E. Giusti, Direct methods in the calculus of variations, World Scientific, 2003. https://doi.org/10.1142/5002
[25]	F. Gmeineder, Partial regularity for symmetric quasiconvex functionals on $BD$, J. Math. Pures Appl., 145 (2021), 83–129. https://doi.org/10.1016/j.matpur.2020.09.005 doi: 10.1016/j.matpur.2020.09.005
[26]	F. Gmeineder, The regularity of minima for the Dirichlet problem on $BD$, Arch. Rational Mech. Anal., 237 (2020), 1099–1171. https://doi.org/10.1007/s00205-020-01507-5 doi: 10.1007/s00205-020-01507-5
[27]	F. Gmeineder, J. Kristensen, Partial regularity for BV minimizers, Arch. Rational Mech. Anal., 232 (2019), 1429–1473. https://doi.org/10.1007/s00205-018-01346-5 doi: 10.1007/s00205-018-01346-5
[28]	F. Gmeineder, J. Kristensen, Quasiconvex functionals of $(p, q)$-growth and the partial regularity of relaxed minimizers, arXiv, 2022. https://doi.org/10.48550/arXiv.2209.01613
[29]	J. Kristensen, On the nonlocality of quasiconvexity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 1–13. https://doi.org/10.1016/S0294-1449(99)80006-7 doi: 10.1016/S0294-1449(99)80006-7
[30]	J. Kristensen, G. Mingione, The singular set of Lipschitzian minima of multiple integrals, Arch. Rational Mech. Anal., 184 (2007), 341–369. https://doi.org/10.1007/s00205-006-0036-2 doi: 10.1007/s00205-006-0036-2
[31]	T. Kuusi, G. Mingione, Guide to nonlinear potential estimates, Bull. Math. Sci., 4 (2014), 1–82. https://doi.org/10.1007/s13373-013-0048-9 doi: 10.1007/s13373-013-0048-9
[32]	T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Rational Mech. Anal., 207 (2013), 215–246. https://doi.org/10.1007/s00205-012-0562-z doi: 10.1007/s00205-012-0562-z
[33]	T. Kuusi, G. Mingione, Partial regularity and potentials, J. Ec. Polytech.-Math., 3 (2016), 309–363. https://doi.org/10.5802/jep.35 doi: 10.5802/jep.35
[34]	T. Kuusi, G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929–1004. https://doi.org/10.4171/jems/780 doi: 10.4171/jems/780
[35]	P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1–28. https://doi.org/10.1007/BF01168345 doi: 10.1007/BF01168345
[36]	P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 391–409. https://doi.org/10.1016/S0294-1449(16)30379-1 doi: 10.1016/S0294-1449(16)30379-1
[37]	P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, In: F. Colombini, A. Marino, L. Modica, S. Spagnolo, Partial differential equations and the calculus of variations, Progress in Nonlinear Differential Equations and Their Applications, Boston: Birkhäuser, 1 (1989), 767–786. https://doi.org/10.1007/978-1-4615-9831-2_11
[38]	C. B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25–53.
[39]	S. Müller, V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math., 157 (2003), 715–742.
[40]	Q. H. Nguyen, N. C. Phuc, A comparison estimate for singular $p$-Laplace equations and its consequences, Arch. Rational Mech. Anal., 247 (2003), 49. https://doi.org/10.1007/s00205-023-01884-7 doi: 10.1007/s00205-023-01884-7
[41]	T. Schmidt, Regularity theorems for degenerate quasiconvex energies with $(p, q)$-growth, Adv. Calc. Var., 1 (2008), 241–270. https://doi.org/10.1515/ACV.2008.010 doi: 10.1515/ACV.2008.010
[42]	T. Schmidt, Regularity of relaxed minimizers of quasiconvex variational integrals with $(p, q)$-growth, Arch. Rational Mech. Anal., 193 (2009), 311–337. https://doi.org/10.1007/s00205-008-0162-0 doi: 10.1007/s00205-008-0162-0

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