April  2019, 12(2): 251-265. doi: 10.3934/dcdss.2019018

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

1. 

Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, I-41125 Modena, Italy

2. 

Dipartimento di Matematica ed Informatica "U. Dini", viale Morgagni 67/a, I-50134 Firenze, Italy

Received  August 2017 Revised  December 2017 Published  August 2018

Fund Project: The authors are members of GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).

We are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let $Ω $ be an open subset of $\mathbb{R}^{n}$. Let $f≤\left( {x, \xi } \right) $ be a real function defined in $Ω × \mathbb{R}^{n}$ satisfying the growth condition $|{f_{\xi x}}\left( {x, \xi } \right)| \le h\left( x \right)|\xi {{\rm{|}}^{p - 1}}$, for $x∈ Ω $ and $\xi ∈ \mathbb{R}^{n}$ with $|\xi {\rm{|}} \ge {M_0}$ for some $M_{0}≥ 0$, with $h \in L_{{\rm{loc}}}^r\left( \Omega \right) $ for some $r>n$. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to $f\left( {x, \xi } \right) $ the so-called natural $p-$growth conditions on the second derivatives ${f_{\xi \xi }}\left( {x, \xi } \right)$; i.e., $\left( {p - 2} \right) - $growth for $|{f_{\xi \xi }}\left( {x, \xi } \right)| $ from above and $\left( {p - 2} \right) - $growth from below for the quadratic form $({f_{\xi \xi }}\left( {x, \xi } \right)\lambda , \lambda {\rm{ }})$; for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer $u \in W_{{\rm{loc}}}^{1, p}\left( \Omega \right) $ of the energy integral $\int_\Omega {f(x, Du\left( x \right)){\mkern 1mu} dx} $.

Citation: Michela Eleuteri, Paolo Marcellini, Elvira Mascolo. Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 251-265. doi: 10.3934/dcdss.2019018
References:
[1]

A. L. BaisónA. ClopR. GiovaJ. Orobitg and A. Passarelli di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal., 46 (2017), 403-430.  doi: 10.1007/s11118-016-9585-7.  Google Scholar

[2]

S. BaraketS. ChebbiN. Chorfi and V. Radulescu, Non-autonomous eigenvalue problems with variable $(p_{1}, p_{2})-$growth, Adv. Nonlinear Stud., 17 (2017), 781-792.  doi: 10.1515/ans-2016-6020.  Google Scholar

[3]

P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., (Special issue in honor of Enzo Mitidieri for his 60th birthday), 121 (2015), 206–222. doi: 10.1016/j.na.2014.11.001.  Google Scholar

[4]

P. Baroni, M. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., (Special issue for N. Ural'tseva), 27 (2016), 347–379. doi: 10.1090/spmj/1392.  Google Scholar

[5]

M. M. BoureanuP. Pucci and V. Radulescu, Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent, Complex Var. Elliptic Equ., 56 (2011), 755-767.  doi: 10.1080/17476931003786709.  Google Scholar

[6]

M. Chipot and 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), 291-303.  doi: 10.1017/S0308210500026378.  Google Scholar

[7]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Rat. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.  Google Scholar

[8]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Rat. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.  Google Scholar

[9]

G. CupiniF. GiannettiR. Giova and A Passarelli di Napoli, Higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients, Nonlinear Anal., 154 (2017), 7-24.  doi: 10.1016/j.na.2016.02.017.  Google Scholar

[10]

G. CupiniM. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with pq growth, Nonlinear Anal., 54 (2003), 591-616.  doi: 10.1016/S0362-546X(03)00087-7.  Google Scholar

[11]

G. CupiniP. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under p, q-growth, Adv. Differential Equations, 19 (2014), 693-724.   Google Scholar

[12]

E. DiBenedetto, $\mathcal{C}^{1+\alpha} $ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., TMA, 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[13]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl., 195 (2016), 1575-1603.  doi: 10.1007/s10231-015-0529-4.  Google Scholar

[14]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz continuity for functionals with variable exponents, Rend. Lincei Mat. Appl., 27 (2016), 61-87.  doi: 10.4171/RLM/723.  Google Scholar

[15]

M. Eleuteri, P. Marcellini and E. Mascolo, Regularity for scalar integrals without structure conditions, Adv. Calc. Var., (2018), in press. doi: 10.1515/acv-2017-0037.  Google Scholar

[16]

M. Eleuteri and A. Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle problems, submitted. Google Scholar

[17]

I. FonsecaN. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations., 7 (2002), 69-95.  doi: 10.1051/cocv:2002004.  Google Scholar

[18]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.  Google Scholar

[19]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York and London, 1968.  Google Scholar

[20]

M. MihailescuP. Pucci and V. Radulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris, 345 (2007), 561-566.  doi: 10.1016/j.crma.2007.10.012.  Google Scholar

[21]

M. MihailescuP. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698.  doi: 10.1016/j.jmaa.2007.09.015.  Google Scholar

[22]

A. Passarelli di Napoli, Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Advances in Calculus of Variations, 7 (2014), 59-89.  doi: 10.1515/acv-2012-0006.  Google Scholar

[23]

V. Radulescu and D. Repovs, Partial Differential Equations with Variable Exponents, Variational methods and qualitative analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. xxi+301 pp. doi: 10.1201/b18601.  Google Scholar

show all references

References:
[1]

A. L. BaisónA. ClopR. GiovaJ. Orobitg and A. Passarelli di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal., 46 (2017), 403-430.  doi: 10.1007/s11118-016-9585-7.  Google Scholar

[2]

S. BaraketS. ChebbiN. Chorfi and V. Radulescu, Non-autonomous eigenvalue problems with variable $(p_{1}, p_{2})-$growth, Adv. Nonlinear Stud., 17 (2017), 781-792.  doi: 10.1515/ans-2016-6020.  Google Scholar

[3]

P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., (Special issue in honor of Enzo Mitidieri for his 60th birthday), 121 (2015), 206–222. doi: 10.1016/j.na.2014.11.001.  Google Scholar

[4]

P. Baroni, M. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., (Special issue for N. Ural'tseva), 27 (2016), 347–379. doi: 10.1090/spmj/1392.  Google Scholar

[5]

M. M. BoureanuP. Pucci and V. Radulescu, Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent, Complex Var. Elliptic Equ., 56 (2011), 755-767.  doi: 10.1080/17476931003786709.  Google Scholar

[6]

M. Chipot and 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), 291-303.  doi: 10.1017/S0308210500026378.  Google Scholar

[7]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Rat. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.  Google Scholar

[8]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Rat. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.  Google Scholar

[9]

G. CupiniF. GiannettiR. Giova and A Passarelli di Napoli, Higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients, Nonlinear Anal., 154 (2017), 7-24.  doi: 10.1016/j.na.2016.02.017.  Google Scholar

[10]

G. CupiniM. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with pq growth, Nonlinear Anal., 54 (2003), 591-616.  doi: 10.1016/S0362-546X(03)00087-7.  Google Scholar

[11]

G. CupiniP. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under p, q-growth, Adv. Differential Equations, 19 (2014), 693-724.   Google Scholar

[12]

E. DiBenedetto, $\mathcal{C}^{1+\alpha} $ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., TMA, 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[13]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl., 195 (2016), 1575-1603.  doi: 10.1007/s10231-015-0529-4.  Google Scholar

[14]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz continuity for functionals with variable exponents, Rend. Lincei Mat. Appl., 27 (2016), 61-87.  doi: 10.4171/RLM/723.  Google Scholar

[15]

M. Eleuteri, P. Marcellini and E. Mascolo, Regularity for scalar integrals without structure conditions, Adv. Calc. Var., (2018), in press. doi: 10.1515/acv-2017-0037.  Google Scholar

[16]

M. Eleuteri and A. Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle problems, submitted. Google Scholar

[17]

I. FonsecaN. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations., 7 (2002), 69-95.  doi: 10.1051/cocv:2002004.  Google Scholar

[18]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.  Google Scholar

[19]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York and London, 1968.  Google Scholar

[20]

M. MihailescuP. Pucci and V. Radulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris, 345 (2007), 561-566.  doi: 10.1016/j.crma.2007.10.012.  Google Scholar

[21]

M. MihailescuP. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698.  doi: 10.1016/j.jmaa.2007.09.015.  Google Scholar

[22]

A. Passarelli di Napoli, Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Advances in Calculus of Variations, 7 (2014), 59-89.  doi: 10.1515/acv-2012-0006.  Google Scholar

[23]

V. Radulescu and D. Repovs, Partial Differential Equations with Variable Exponents, Variational methods and qualitative analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. xxi+301 pp. doi: 10.1201/b18601.  Google Scholar

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