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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[1]

Arnulf Jentzen, Benno Kuckuck, Thomas Müller-Gronbach, Larisa Yaroslavtseva. Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3707-3724. doi: 10.3934/dcdsb.2021203

[2]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163

[3]

Antônio Luiz Pereira, Severino Horácio da Silva. Continuity of global attractors for a class of non local evolution equations. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1073-1100. doi: 10.3934/dcds.2010.26.1073

[4]

Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741

[5]

Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007

[6]

Jeongmin Han. Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2617-2640. doi: 10.3934/cpaa.2020114

[7]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417

[8]

Sami Aouaoui. On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1767-1784. doi: 10.3934/cpaa.2017086

[9]

Vladimír Špitalský. Local correlation entropy. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249

[10]

Dashun Xu, Z. Feng. A metapopulation model with local competitions. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 495-510. doi: 10.3934/dcdsb.2009.12.495

[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505

[13]

Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261

[14]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

[15]

André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 2022, 11 (3) : 749-779. doi: 10.3934/eect.2021024

[16]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294

[17]

Roland Martin. On simple Igusa local zeta functions. Electronic Research Announcements, 1995, 1: 108-111.

[18]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic and Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205

[19]

Alberto A. Pinto, João P. Almeida, Telmo Parreira. Local market structure in a Hotelling town. Journal of Dynamics and Games, 2016, 3 (1) : 75-100. doi: 10.3934/jdg.2016004

[20]

Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (288)
  • HTML views (169)
  • Cited by (1)

[Back to Top]