Regularity Results for Local Minimizers of Functionals with Discontinuous Coefficients

Raffaella Giova, Antonia Passarelli di Napoli

Abstract


We give an overview on recent regularity results of local vectorial minimizers of under two main features: the energy density is uniformly convex with respect to the gradient variable only at infinity and it depends on the spatial variable through a possibly discontinuous coefficient. More precisely, the results that we present tell that a suitable weak differentiability property of the integrand as function of the spatial variable implies the higher differentiability and the higher integrability of the gradient of the local minimizers. We also discuss the regularity of the local solutions of nonlinear elliptic equations under a fractional Sobolev assumption.


Keywords


Regularity; Asymptotically convex; Minimizer

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References


E. Acerbi, N. Fusco: Partial regularity under anisotropic (p; q) growth conditions, J. Differential Equations 107 (1994), no. 1, 46-67.

A. Baison, A. Clop, J. Orobitg: Beltrami equations with coefficient in the fractional Sobolev space Wᶱ; n/ᶱ. Preprint arXiv:1507.05799 (2015)

A. Baison, A. Clop, R. Giova, J. Orobitg, A. Passarelli di Napoli: Fractional differentiability for solutions of nonlinear elliptic equations, Potential Analysis DOI 10.1007/s11118-016-9585-7.

L. Boccardo, P. Marcellini, C. Sbordone: L∞-regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A 4 (1990) 219-225.

M. Carozza, J. Kristensen, A. Passarelli di Napoli. Higher differentiability of minimizers of convex variational integrals, Annales Inst. H. Poincaré (C) Non Linear Analysis , 28 (2011) (3), 395–411.

P. Celada, G. Cupini, M. Guidorzi: Existence and regularity of minimizers of nonconvex integrals with p-q growth, ESAIM Control Optim. Calc. Var. 13 (2007), 343-358.

M. Chipot, L.C. Evans: Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3-4, 291-303.

A. Clop, D. Faraco, J. Mateu, J. Orobitg, X. Zhong. Beltrami equations with coefficient in the Sobolev Space W1;p, Publ. Mat. 53 (2009), 197-230.

A. Clop, R. Giova, A. Passarelli di Napoli: Extra fractional differentiability for solutions of p-harmonic equations, Preprint (2016).

M. Colombo, G. Mingione: Regularity for double phase variational problems, Arch. Rat. Mech. Anal. 215 (2015) 443-496.

M. Colombo, G. Mingione: Bounded minimisers of double phase variational integrals, Arch. Rat. Mech. Anal. 218 (2015), 219-273.

W. Cruz, J. Mateu, J. Orobitg: Beltrami equation with coefficient in Sobolev and Besov spaces, Canad. J. Math. 65 (2013), 1217-1235.

D. Cruz-Uribe, K. Moen, S. Rodney: Regularity results for weak solutions of elliptic PDE’s below the natural exponent, Ann. Mat. Pura Appl. 195 (2016), n.3, 725-740.

G. Cupini, N. Fusco, R. Petti: Hölder continuity of local minimizers, J. Math. Anal. Appl. 235

(1999) 578-597.

G. Cupini, F. Giannetti, R. Giova, A. Passarelli di Napoli: Higher integrability estimates for minimizers of asymptotically convex integrals with discontinuous coefficients. Nonlinear Anal. (2016) doi.org/10.1016/j.na.2016.02.017.

G. Cupini, F. Giannetti, R. Giova, A. Passarelli di Napoli: Regularity for minimizers of integrals with non standard growth conditions and discontinuous coefficients. Preprint 2016.

G. Cupini, M. Guidorzi, E. Mascolo: Regularity of minimizers of vectorial integrals with p - q growth, Nonlinear Anal.54 (2003) 591-616.

G. Cupini, P. Marcellini, E. Mascolo: Local boundedness of solutions to quasilinear elliptic systems. Manuscripta Math. 137, 287-315 (2012)

G. Cupini, P. Marcellini, E. Mascolo: Local boundedness of solutions to some anisotropic elliptic systems. Recent trends in nonlinear partial differential equations. II. Stationary problems, 169-186, Contemp. Math., 595, Amer. Math. Soc., Providence, RI, 2013.

G. Cupini, P. Marcellini, E. Mascolo: Regularity of minimizers under limit growth conditions, Nonlinear Anal. (2016), doi.org/10.1016/j.na.2016.06.002.

G. Cupini, A.P. Migliorini: Hölder continuity for local minimizers of a nonconvex variational problem, J. Convex Anal. 10 (2003) 389-408.

B. Dacorogna: Direct methods in the Calculus of Variations, 2nd edition. Appl. Math. Sci. 78, Springer, New York, 2008.

A. Dall’Aglio, E. Mascolo: L∞ estimates for a class of nonlinear elliptic systems with nonstandard

growth. Atti Sem. Mat. Fis. Univ. Modena 50 (2002), no. 1, 6583

A. Dall’Aglio, E. Mascolo, G. Papi: Local boundedness for minima of functionals with nonstandard growth conditions, Rend. Mat. Appl.18 (1998) 305-326.

E. De Giorgi: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Unione Mat. Ital. IV. Ser. 1 (1968) 135-137.

M. Eleuteri, P. Marcellini, E. Mascolo: Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura e Appl., 195 (2016) 1575-1603.

L. Esposito, F. Leonetti, G. Mingione: Regularity results for minimizers of irregular integrals with (p; q) growth, Forum Mathematicum 14 (2002) 245-272.

L. Esposito, F. Leonetti, G. Mingione: Sharp regularity for functionals with (p,q) growth, J. Differential Equations 204 (2004) 5-55.

I. Fonseca, N. Fusco, P. Marcellini: An existence result for a nonconvex variational problem via regularity, ESAIM: Control, Optim. Calc. Var. 7 (2002) 69-96.

I. Fonseca, I. Mal´y, G. Mingione: Scalar minimizers with fractal singular sets, Arch. Rat. Mech. Anal. 172 (2004) 295-307.

M. Foss, A. Passarelli di Napoli, A. Verde: Morrey regularity and continuity results for almost minimizers of asymptotically convex integrals, Appl. Math. (Warsaw) 35 (2008), 335-353.

M. Foss, A. Passarelli di Napoli, A. Verde: Global Morrey regularity results for asymptotically convex variational problems, Forum Math. 20 (2008), 921-953.

M. Foss, A. Passarelli di Napoli, A. Verde: Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems, Ann. Mat. Pura Appl. (4) 189 (2010), 127-162.

N. Fusco, C. Sbordone: Local boundedness of minimizers in a limit case, Manuscripta Mathematica 69, (1990) 19-25.

N. Fusco, C. Sbordone: Some remarks on the regularity of minima of anisotropic integrals, Commun. in Partial Diff. Equ. 18 (1993) 153-167.

F. Giannetti: A C1;α partial regularity result for integral functionals with p(x)- growth condition, Adv. Calc. Var., doi: 10.1515/acv-2015-0011.

F. Giannetti, A. Passarelli di Napoli: Hölder continuity of degenerate p-harmonic functions, Ann. Acad. Sci. Fennicae Math. 39 (2014) 567-577.

F. Giannetti, A. Passarelli di Napoli: Higher differentiability of minimizers of variational integrals with variable exponents. Math. Zeitschrift 280 (2015),n.3, 873892.

F. Giannetti, A. Passarelli di Napoli, C. Scheven: Higher differentiability of solutions of parabolic systems with discontinuous coefficients, J. London Math. Soc. doi:10.1112/jlms/jdw019.

M. Giaquinta. Growth conditions and regularity: a counterexample, Manuscripta Math. 59 (1987) 245–248.

M. Giaquinta, G. Modica: Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math. 57 (1986) 55-99.

R. Giova: Higher differentiability for n-harmonic systems with Sobolev coefficients, J. Diff. Equations 259 (2015), 5667-5687.

R. Giova: Regularity results for non autonomous functionals with L log L-growth and Orlicz Sobolev coefficients, Nonlinear Differential Equations and Applications (to appear)

R. Giova, A. Passarelli di Napoli: Regularity results for a priori bounded minimizers of non autonomous functionals with discontinuous coefficients - Preprint (2016)

E. Giusti: Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

P. Haijlasz: Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996) 403-415.

D. Haroske. Envelopes and Sharp Embeddings of Function Spaces, Chapman and Hall CRC (2006).

T. Iwaniec, C. Sbordone: Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994) 143-162.

T. Iwaniec, C. Sbordone: Riesz transforms and elliptic PDEs with VMO coefficients, J. Analyse. Math. 74 (1998) 183-212.

J. Kinnunen, S. Zhou: A local estimate for nonlinear equations with discontinuous coefficients. Comm. Part. Diff. Equ. 24 (1999), 2043-2068.

J. Kristensen, G. Mingione: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180 (2006) 331-398.

J. Kristensen, G. Mingione: Boundary regularity in variational problems, Arch. Ration. Mech. Anal. 198 (2010) 369-455.

T. Kuusi, G. Mingione: Universal potential estimates, J. Funct. Anal. 262 (2012) 4205-4269.

C. Leone, A. Passarelli di Napoli, A. Verde: Lipschitz regularity for some asymptotically subquadratic problems. Nonlinear Anal. 67 (2007), 1532-1539.

P. Marcellini: Un example de solution discontinue d’un problème variationnel dans le cas scalaire, Preprint 11, Istituto Matematico “U.Dini”, Università di Firenze, 1987.

P. Marcellini: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Rat. Mech. Anal. 105 (1989) 267-284.

P. Marcellini: Regularity and existence of solutions of elliptic equations with p; q-growth conditions, J. Differential Equations 90 (1991) 1-30.

P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993) 296–333.

P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa 23 (1996) 1-25.

G. Mingione: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51 (2006) 355-426.

A. Passarelli di Napoli: Higher differentiability of minimizers of variational integrals with Sobolev coefficients. Adv. Cal. Var. 7 (1) (2014), 59-89.

A. Passarelli di Napoli: Higher differentiability of solutions of elliptic systems with Sobolev coefficients: the case p = n = 2. Pot. Anal. 41 (3) (2014), 715–735

A. Passarelli di Napoli: Regularity results for non-autonomous variational integrals with discontinuous coefficients. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (4) (2015), 475-496.

A. Passarelli di Napoli, A. Verde: A regularity result for asymptotically convex problems with lower order terms. J. Convex Anal. 15 (2008), 131-148.

V. Sver`ak, X. Yan: Non-Lipschitz minimizers of smooth uniformly convex functionals Proc. Natl. Acad. Sci. USA 99 (24) (2002) 15269-15276.

H.Triebel. Theory of function spaces. Monographs in Mathematics. 78. Birkhauser, Basel, 1983.




DOI: 10.6092/issn.2240-2829/6695

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