On renormalized solutions to elliptic inclusions with nonstandard growth

Abstract

We study the elliptic inclusion given in the following divergence form

$$\begin{aligned}&-\mathrm {div}\,A(x,\nabla u) \ni f\quad \mathrm {in}\quad \Omega ,\\&u=0\quad \mathrm {on}\quad \partial \Omega . \end{aligned}$$

As we assume that \(f\in L^1(\Omega )\), the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction A which necessitates the use of the nonseparable and nonreflexive Musielak–Orlicz spaces. We prove the existence and uniqueness of the renormalized solution as well as, under additional assumptions on the problem data, its boundedness. The key difficulty, the lack of a Carathéodory selection of the maximally monotone multifunction is overcome with the use of the Minty transform.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Ahmida, Y., Chlebicka, I., Gwiazda, P., Youssfi, A.: Gossez’s approximation theorems in the Musielak–Orlicz–Sobolev spaces. J. Funct. Anal. 275, 2538–2571 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alberico, A., Chlebicka, I., Cianchi, A., Zatorska-Goldstein, A.: Fully anisotropic elliptic problems with minimally integrable data. Calc. Var. PDEs 58, 186 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Alberico, A., Cianchi, A.: Comparison estimates in anisotropic variational problems. Manuscr. Math. 126, 481–503 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alberti, G., Ambrosio, L.: A geometrical approach to monotone functions in \({\mathbb{R}}^{n}\). Math. Z. 230, 259–316 (1999)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability measures. Birkhäuser, Basel (2005)

    Google Scholar 

  6. 6.

    Ball, J.M.: A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions. In: Proceedings of an NSF-CNRS Joint Seminar Held in Nice, France, January 18–22, 1988, pp. 207–215 (1989)

  7. 7.

    Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107, 655–663 (1989)

    MathSciNet  Google Scholar 

  8. 8.

    Baroni, P., Colombo, M., Mingione, G.: Nonautonomous functionals, borderline cases and related function classes. St. Petersburg Math. J. 27, 347–379 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. Partial Differ. Equ. 57, 57–62 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Bendahmane, M., Wittbold, P.: Renormalized solutions for nonlinear elliptic equations with variable exponents and \(L^1\) data. Nonlinear Anal. Theory Methods Appl. 70, 567–583 (2009)

    Article  MathSciNet  Google Scholar 

  11. 11.

    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^{1}\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 22, 241–273 (1995)

    MathSciNet  Google Scholar 

  12. 12.

    Benilan, P., Wittbold, P.: On mild and weak solutions of elliptic-parabolic problems. Adv. Differ. Equ. 1, 1053–1073 (1996)

    MathSciNet  Google Scholar 

  13. 13.

    Bhattacharya, T., Leonetti, F.: A new poincaré inequality and its application to the regularity of minimizers of integral functionals with nonstandard growth. Nonlinear Anal. 17, 833–839 (1991)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17, 641–655 (1992)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Buliíček, M., Diening, L., Schwarzacher, S.: Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems. Anal. PDE 9, 1115–1151 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Bulíčcek, M., Gwiazda, P., Kalousek, M., Świerczewska Gwiazda, A.: Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces. Nonlinearity 32, 1073 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Bulíčcek, M., Gwiazda, P., Malek, J., Świerczewska Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44, 2756–2801 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Bulíčcek, M., Gwiazda, P., Świerczewska Gwiazda, A.: On unified theory for scalar conservation laws with fluxes and sources discontinuous with respect to the unknown. J. Differ. Equ. 262, 313–364 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Byun, S.-S., Oh, J.: Regularity results for generalized double phase functionals. Anal. PDE (to appear)

  20. 20.

    Byun, S.-S., Youn, Y.: Potential estimates for elliptic systems with subquadratic growth. J. Math. Pures Appl. 131, 193–224 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Carillo, J.: Conservation laws with discontinuous flux functions and boundary condition. J. Evol. Equ. 3, 283–301 (2003)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Chiado’Piat, V., Dal Maso, G., Defrancheschi, A.: G-convergence of monotone operators. Annales de l’H.P., Sect. C 7, 123–160 (1990)

    MathSciNet  Google Scholar 

  23. 23.

    Chlebicka, I.: Gradient estimates for problems with Orlicz growth. Nonlinear Anal. 194, 111364 (2020)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Chlebicka, I.: A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces. Nonlinear Anal. 175, 1–27 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Chlebicka, I., De Filippis, C.: Removable sets in non-uniformly elliptic problems. Annali di Matematica 199, 619–649 (2020)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Chlebicka, I., Giannetti, F., Zatorska-Goldstein, A.: Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or \(l^1\) data. J. Math. Anal. Appl. 479, 185–213 (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Chlebicka, I., Gwiazda, P., Zatorska-Goldstein, A.: Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev’s phenomenon. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 36, 1431–1465 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Chlebicka, I., Gwiazda, P., Zatorska-Goldstein, A.: Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev’s phenomenon. J. Differ. Equ. 267, 1129–1166 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Chlebicka, I., Gwiazda, P., Zatorska-Goldstein, A.: Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions. J. Differ. Equ. 265, 5716–5766 (2018)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Cianchi, A.: A fully anisotropic Sobolev inequality. Pac. J. Math. 196, 283–294 (2000)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Cianchi, A.: Symmetrization in anisotropic elliptic problems. Commun. Partial Differ. Equ. 32, 693–717 (2007)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Cianchi, A., Maz’ya, V.: Quasilinear elliptic problems with general growth and merely integrable, or measure, data. Nonlinear Anal. 164, 189–215 (2017)

    MathSciNet  Article  Google Scholar 

  33. 33.

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

    MathSciNet  Article  Google Scholar 

  34. 34.

    De Filippis, C., Mingione, G.: Manifold constrained non-uniformly elliptic problems. J. Geom. Anal. 30, 1661–1723 (2020)

    MathSciNet  Article  Google Scholar 

  35. 35.

    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (2015)

    Google Scholar 

  37. 37.

    Francfort, G., Murat, F., Tartar, L.: Monotone operators in divergence form with x-dependent multivalued graphs. Bollettino dell’Unione Matematica Italiana 7B, 23–59 (2004)

    MathSciNet  Google Scholar 

  38. 38.

    Griewank, A., Rabier, P.J.: On the smoothness of convex envelopes. Trans. Am. Math. Soc. 322, 691–709 (1990)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Gwiazda, P., Minakowski, P., Wróblewska-Kamińska, A.: Elliptic problems in generalized Orlicz–Musielak spaces. Centr. Eur. J. Math. 10, 2019–2032 (2012)

    MathSciNet  Google Scholar 

  40. 40.

    Gwiazda, P., Skrzypczak, I., Zatorska-Goldstein, A.: Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space. J. Differ. Equ. 264, 341–377 (2018)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Gwiazda, P., Świerczewska Gwiazda, A.: On non-Newtonian fluids with a property of rapid thickening under different stimulus. Math. Models Methods Appl. Sci. 18, 1073–1092 (2008)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Gwiazda, P., Świerczewska Gwiazda, A., Wittbold, P., Zimmerman, A.: Multi-dimensional scalar balance laws with discontinuous flux. J. Funct. Anal. 267, 2846–2883 (2014)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Gwiazda, P., Świerczewska Gwiazda, A., Wróblewska, A.: Monotonicity methods in generalized Orlicz spaces for a class on non-Newtonian fluids. Math. Methods Appl. Sci. 33, 125–137 (2010)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Gwiazda, P., Wittbold, P.. Wróblewska, A., Zimmermann, A.: Corrigendum to ”Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces” [J. Differential Equations] 253(2), 635-666 (2012). J. Differ. Equ. 253, 2734–2738 (2012)

  45. 45.

    Gwiazda, P., Wittbold, P., Wróblewska, A., Zimmermann, A.: Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. J. Differ. Equ. 253, 635–666 (2012)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Gwiazda, P., Wittbold, P., Wróblewska-Kamińska, A., Zimmermann, A.: Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces. Nonlinear Anal. 129, 1–36 (2015)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Gwiazda, P., Zatorska-Goldstein, A.: On elliptic and parabolic systems with x-dependent multivalued graphs. Math. Methods Appl. Sci. 30, 213–236 (2007)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Harjulehto, P., Hástö, P.: Orlicz Spaces and Generalized Orlicz Spaces. Lecture Notes in Mathematics, vol. 2236. Springer, Berlin (2019)

    Google Scholar 

  49. 49.

    Harjulehto, P., Hástö, P., Lee, M.: Hölder continuity of quasiminimizers and \(\omega \)-minimizers of functionals with generalized orlicz growth. Ann. Sc. Norm. Super. Pisa Cl. Sci. to appear (2019)

  50. 50.

    Harjulehto, P., Hástö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. Partial Differ. Equ. 56, 22 (2017)

    Article  Google Scholar 

  51. 51.

    Hástö, P., Ok, J.: Calderón–Zygmund estimates in generalized Orlicz spaces. J. Differ. Equ. 267, 2792–2823 (2019)

    Article  Google Scholar 

  52. 52.

    Kilpeláinen, T., Kuusi, T., Tuhola-Kujanpää, A.: Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 775–795 (2011)

    MathSciNet  Article  Google Scholar 

  53. 53.

    Lavrentiev, M.: Sur quelques problemes du calcul des variations. Ann. Mat. Pura Appl. 41, 107–124 (1927)

    MathSciNet  Google Scholar 

  54. 54.

    Müller, S.: Variational Models for Microstructure and Phase Transitions. Lecture Notes, vol. 2. Max Planck Institut für Mathematik in den Naturwissenschaften, Leipzig (1998)

    Google Scholar 

  55. 55.

    Pedregal, P.: Parametrized Measures and Variational Principles. Progress in Nonlinear Differential Equations and Their Applications, vol. 30. Springer, Basel (1997)

    Google Scholar 

  56. 56.

    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Dordrecht (2009)

    Google Scholar 

  57. 57.

    Schappacher, G.: A notion of Orlicz spaces for vector values functions. Appl. Math. 50, 355–386 (2005)

    MathSciNet  Article  Google Scholar 

  58. 58.

    Skaff, M.S.: Vector valued Orlicz spaces. II. Pac. J. Math. 28, 413–430 (1969)

    MathSciNet  Article  Google Scholar 

  59. 59.

    Świerczewska Gwiazda, A.: Anisotropic parabolic problems with slowly or rapidly growing terms. Colloq. Math. 134, 113–130 (2014)

    MathSciNet  Article  Google Scholar 

  60. 60.

    Świerczewska Gwiazda, A.: Nonlinear parabolic problems in Musielak–Orlicz spaces. Nonlinear Anal. 98, 48–65 (2014)

    MathSciNet  Article  Google Scholar 

  61. 61.

    Talenti, G.: Boundedness of minimizers. Hokkaido Math. J. 19, 259–279 (1990)

    MathSciNet  Article  Google Scholar 

  62. 62.

    Trudinger, N.S.: An imbedding theorem for \(h_0(g, \omega )\) spaces. Studia Math. 50, 17–30 (1974)

    MathSciNet  Article  Google Scholar 

  63. 63.

    Wittbold, P., Zimmermann, A.: Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and \(L^{1}\)-data. Nonlinear Anal. Theory Methods Appl. 72, 2990–3008 (2010)

    Article  MathSciNet  Google Scholar 

  64. 64.

    Wróblewska, A.: Steady flow of non-Newtonian fluids—monotonicity methods in generalized Orlicz spaces. Nonlinear Anal. 72, 4136–4147 (2010)

    MathSciNet  Article  Google Scholar 

  65. 65.

    Zhang, C., Zhou, S.: Entropy and renormalized solutions for the p(x)-laplacian equation with measure data. Bull. Aust. Math. Soc. 82, 459–479 (2010)

    MathSciNet  Article  Google Scholar 

  66. 66.

    Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)

    MathSciNet  Google Scholar 

  67. 67.

    Zhikov, V.V.: On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. (N.Y.) 173, 463–570 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Piotr Kalita.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Work of AD has been supported by the Cracow University of Economics by the Project No.36/EIM/2020/POT. Work of PG has been supported by the National Science Center of the Republic of Poland by the Project No. UMO-2018/31/B/ST1/02289. Work of PK has been supported by the National Science Center of the Republic of Poland by the Project UMO-2016/22/A/ST1/00077. We thank Iwona Chlebicka for discussion and useful remarks. We also thank anonymous reviewers for valuable comments.

Communicated by Y. Giga.

Appendices

Appendix A: N-functions and Musielak–Orlicz spaces

N-functions. We start from the definition of N-functions.

Definition A.1

The function \(M:\Omega \times {\mathbb {R}}^d \rightarrow [0,\infty )\) is an N-function if

  1. (N1)

    M is Carathéodory, that is, \(M(\cdot ,\xi )\) is measurable for every \(\xi \in {\mathbb {R}}^d\) and \(M(x,\cdot )\) is continuous for almost every \(x\in \Omega \),

  2. (N2)

    \(M(x,\xi ) = M(x,-\xi )\) for every \(\xi \in {\mathbb {R}}^d\) a.e. in \(\Omega \) and \(M(x,\xi ) = 0\) is and only if \(\xi =0\) a.e. in \(\Omega \),

  3. (N3)

    \(M(x,\cdot )\) is convex for almost every \(x\in \Omega \),

  4. (N4)

    M has superlinear growth in \(\xi \) at zero and infinity, that is,

    $$\begin{aligned} \lim _{|\xi |\rightarrow 0}\mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{M(x,\xi )}{|\xi |} = 0\qquad \text {and}\qquad \lim _{|\xi |\rightarrow \infty }\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{M(x,\xi )}{|\xi |} = \infty . \end{aligned}$$
  5. (N5)

    \( \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\inf _{|\xi |=s}M(x,\xi ) > 0 \ \text {for every}\ s\in (0,\infty )\) and \({{\,\mathrm{ess\,sup}\,}}_{x\in \Omega }M(x,\xi ) < \infty \ \text {for every}\ \xi \ne 0. \)

If \(d=1\) we will use small letters, such as m, to denote N-functions, we will call such functions one dimensional N-functions. In such case by assumption (N2) there holds \(m(x,-\xi ) = m(x,\xi )\) for every \(\xi \in {\mathbb {R}}\) so it is enough to define one dimensional N-function for \(\xi \in [0,\infty )\). Indeed we will assume that one dimensional N-functions are defined only on \(\Omega \times [0,\infty )\), and with some abuse of notation we will define one dimensional N-functions sometimes on the whole line and sometimes on the half-line. If an N-function does not depend on x we will call it homogeneous. So, homogeneous N-function leads from \({\mathbb {R}}^d\) to \( [0,\infty )\) and homogeneous one dimensional N-function from \([0,\infty )\) to \([0,\infty )\). If \(M:\Omega \times {\mathbb {R}}^d\rightarrow {\mathbb {R}}\) then its complementary function \({\widetilde{M}}\) is defined by the Fenchel transform in the following way

$$\begin{aligned} {\widetilde{M}}(x,\eta ) = \sup _{\xi \in {\mathbb {R}}^d} \left\{ \xi \cdot \eta - M (x,\xi ) \right\} . \end{aligned}$$
(A.1)

In the following results we discuss the assumptions in the definition of an N-function and establish some of its properties. We remark that the behavior of an N-function close to the origin is not important for the main results of the present paper, as it does not influence the generated Musielak–Orlicz space nor the arguments of the proofs. In the discussion below, however, for the sake of the exposition completeness, we discuss both the behavior close to infinity as well as close to the origin.

Remark A.2

It is easy to verify that the complementary function of a function which satisfies (N1)–(N4) satisfies (N1)–(N3). It does not have to satisfy either the first, or the second assertion of (N4). Indeed, let \(\Omega =(0,1)\) and let

$$\begin{aligned} M(x,\xi ) = {\left\{ \begin{array}{ll} x\frac{|\xi |^2}{2}\ \text {when}\ |\xi |\le 1,\\ \frac{|\xi |^2}{2}-\frac{1}{2}+\frac{x}{2}\ \text {otherwise}. \end{array}\right. }\quad \text {Then}\quad {\widetilde{M}}(x,\eta )={\left\{ \begin{array}{ll}\frac{|\eta |^2}{2x}\ \text {when}\ |\eta |\le x,\\ |\eta |-\frac{x}{2}\ \text {when}\ |\eta |\in (x,1),\\ \frac{|\eta |^2}{2}-\frac{x}{2}+\frac{1}{2}\ \text {when}\ |\eta |>1. \end{array}\right. }. \end{aligned}$$

It is clear that

$$\begin{aligned} \lim _{|\xi |\rightarrow 0}\mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\eta )}{|\eta |} = \frac{1}{2}. \end{aligned}$$

Moreover,

$$\begin{aligned}&\text {if}\ \ M(x,\xi ) = {\left\{ \begin{array}{ll} \frac{|\xi |^2}{2}\quad \text {when}\quad |\xi |\le \sqrt{2},\\ \frac{|\xi |^2}{2x} + 1 - \frac{1}{x}\quad \text {otherwise}, \end{array}\right. }\ \ \text {then}\\&{\widetilde{M}}(x,\eta ) = {\left\{ \begin{array}{ll} \frac{|\eta |^2}{2}\quad \text {when}\quad |\eta |\le \sqrt{2},\\ \sqrt{2}|\eta | - 1\quad \text {when}\quad \eta \in (\sqrt{2},\sqrt{2}/x),\\ \frac{|\eta |^2x}{2} - 1 + \frac{1}{x}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

It is not hard to verify that

$$\begin{aligned} \lim _{|\eta |\rightarrow \infty }\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\eta )}{|\eta |} = \sqrt{2}. \end{aligned}$$

Note that the two examples do not satisfy (N5). As we will later show, the complementary function of an N-function is also an N-function.

In the following Lemma A.3 and Remark A.4 we establish that N-functions always have a minorant and majorant being one dimensional homogeneous N-functions.

Lemma A.3

Let \(M:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) be a function satisfying (N1)–(N4). Then \(M:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) is an N-function (i.e. it satisfies (N5)) if and only if it is stable, i.e., if there exist homogeneous one dimensional N-functions \(m_1, m_2:[0,\infty )\rightarrow [0,\infty )\) such that for every \(\xi \in {\mathbb {R}}^d\) and almost every \(x\in \Omega \) there holds

$$\begin{aligned} m_1(|\xi |) \le M(x,\xi ) \le m_2(|\xi |). \end{aligned}$$

In particular every N-function is stable.

Proof

The fact that stability implies (N5) is straightforward. For the opposite implication define \(m_2:[0,\infty ) \rightarrow [0,\infty )\) by the formula

$$\begin{aligned} m_2(s) = \mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega } \sup _{|\xi |=s} M(x,\xi ). \end{aligned}$$

It is straightforward to check that this function is finite, nonzero for \(\xi \ne 0\), and satisfies (N1)–(N4). To get the lower bound, let us define

$$\begin{aligned} m_{inf}(s) = \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } \inf _{|\xi |=s} M(x,\xi ). \end{aligned}$$

The function \(m_{inf}\) is nondecreasing and hence it has at most countable number of discontinuities and each of these discontinuities is a jump point. Define

$$\begin{aligned} m_{lsc}(s) = {\left\{ \begin{array}{ll} m_{inf}(s)\ \mathrm {if}\ m_{inf}\ \text {is continuous at}\ s,\\ m_{inf}(s^-)\ \mathrm {otherwise}. \end{array}\right. } \end{aligned}$$

This function is in fact the lower semicontinuous envelope of \(m_{inf}\). Now define \(m_1:[0,\infty ) \rightarrow [0,\infty )\) as

$$\begin{aligned} m_1(s) = \widetilde{\widetilde{m_{lsc}}}(s). \end{aligned}$$

that is, the greatest convex minorant of \(m_{lsc}\). The fact that \(m_1\) satisfies (N1)–(N3), (N5), as well as the growth at zero in (N4) is clear. To prove the growth at infinity assume, for contradiction, that there exist constants \(c< \infty \), \(R > 0\) and \(\epsilon >0\) such that for every \(s\ge R\)

$$\begin{aligned} m_1(s) < cs +\epsilon . \end{aligned}$$

So, for a sequence \(s_n\rightarrow \infty \), there exist numbers \(s_n^1,s_n^2\) and \(\lambda _n^1, \lambda _n^2\ge 0\) such that \(\lambda _n^1 + \lambda _n^2 = 1\) and

$$\begin{aligned} (s_n,m_1(s_n)) = \left( \lambda _n^1 s_n^1 + \lambda _n^1 s_n^2, \lambda _n^1 m_{lsc}(s_n^1) + \lambda _n^1 m_{lsc}(s_n^2)\right) . \end{aligned}$$

Moreover

$$\begin{aligned} s_n^1 \ge s_n, \quad m_{lsc}(s_n^1) = m_1(s_n^1)\quad \text {and}\quad \lambda _n^1 > 0. \end{aligned}$$

Such a choice of \(s_n^1, \lambda _n^1, s_n^2, \lambda _n^2\) is possible due to [38, Theorem 2.1, Remark 2.1]. This means that \(s_n^1\rightarrow \infty \) as \(n\rightarrow \infty \) and \(m_{lsc}(s_n^1) = m_1(s_n^1) < cs_n^1 +\epsilon \), whence

$$\begin{aligned} \frac{m_{lsc}(s_n^1)}{s_n^1} \le c + \frac{\epsilon }{s_n^1}. \end{aligned}$$

Thus we can construct a sequence \(r_n \rightarrow \infty \) such that

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{m_{inf}(r_n)}{r_n}\le c. \end{aligned}$$

This is a contradiction with superlinear growth at infinity of M. \(\square \)

Remark A.4

We are tempted to replace the lower bound in (N5) with its weaker version

$$\begin{aligned} \mathop {{\mathrm{\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits }}}\limits _{x\in \Omega }M(x,\xi ) > 0\ \text {for every}\ \xi \ne 0. \end{aligned}$$

Unfortunately in such case Lemma A.3 does not hold anymore. To demonstrate this consider \(\Omega =(0,1)\), \(d=2\) and let \(\theta (\xi )\) denote the angular polar coordinate of \(\xi \). Define the function

$$\begin{aligned} P(x,\xi ) = {\left\{ \begin{array}{ll} |\xi |^2\ \ \text {if}\ \ |\xi |\ne 1,\\ 1\ \ \text {if}\ \ |\xi |= 1\ \text {and}\ \theta (\xi )\notin (0,x)\cup (\pi ,\pi +x),\\ x+(1-x)\cos ^2\left( \frac{\theta (\xi )}{x}\pi \right) \ \text {if}\ \ |\xi |= 1\ \text {and} \ \theta (\xi )\in (0,x),\\ x+(1-x)\cos ^2\left( \frac{\theta (\xi )-\pi }{x}\pi \right) \ \text {if}\ \ |\xi |= 1\ \text {and} \ \theta (\xi )\in (\pi ,\pi +x). \end{array}\right. } \end{aligned}$$

Note that P is lower semicontinuous with respect to \(\xi \). Define \(M(x,\xi ) = \widetilde{{\widetilde{P}}}(x,\xi )\), the convex envelope of P. Then if only \(|\xi |=1\) and \(\theta (\xi )\in (0,1)\cup (\pi ,\pi +1)\) we obtain

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } M(x,\xi ) \le \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } P(x,\xi ) = \frac{\theta (\xi )}{2}. \end{aligned}$$

This means that

$$\begin{aligned} \inf _{|\xi |=1}\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } M(x,\xi ) = 0, \end{aligned}$$

and hence we can have only \(m_1(1) = 0\). Lower semicontinuity of P with respect to \(\xi \) and [38, Theorem 2.1, Remark 2.1] imply that \({{\,\mathrm{ess\,inf}\,}}_{x\in \Omega } M(x,\xi ) > 0\) for every nonzero \(\xi \). We will verify that M satisfies all remaining conditions in the definition on an N-function. Indeed, (N1)-(N3), growth at zero in (N4), and condition with \({{\,\mathrm{ess\,sup}\,}}\) in (N5) are clear. We will verify growth at infinity in (N4). Assume for contradiction that there exists a sequence \(\xi _n\) with \(|\xi _n|\rightarrow \infty \) and a constant \(C>0\) such that

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{M(x,\xi _n)}{|\xi _n|} \le C. \end{aligned}$$

This means that there exists \(x_n \in (0,1)\) such that

$$\begin{aligned} \frac{M(x_n,\xi _n)}{|\xi _n|} \le C+1. \end{aligned}$$

Now [38, Theorem 2.1] implies that there exist nonnegative numbers \(\lambda ^1_n+\lambda ^2_n+\lambda ^3_n = 1\) and \(\xi ^1_n,\xi ^2_n,\xi ^3_n\in {\mathbb {R}}^2\) such that \(\xi _n = \lambda ^1_n\xi ^1_n + \lambda ^2_n\xi ^2_n+\lambda ^3_n\xi ^3_n\) and

$$\begin{aligned} M(x_n,\xi _n) = \lambda ^1_nP(x_n,\xi ^1_n)+\lambda ^2_nP(x_n,\xi ^2_n)+\lambda ^3_nP(x_n,\xi ^3_n). \end{aligned}$$

We can assume that \(|\xi ^1_n|\ge |\xi _n|\) and \(\lambda ^1_n > 0\). If, for a given n, neither of \(|\xi ^k_n|\) is on a unit circle then

$$\begin{aligned} |\xi _n|^2\le \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3|\xi _n^3|^2 = M(x_n,\xi _n) \le P(x_n,\xi _n)= |\xi _n|^2. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{M(x_n,\xi _n)}{|\xi _n|} = |\xi _n|\le C+1. \end{aligned}$$

If exactly one of \(\xi _n^k\) is on the unit circle, say \(\xi ^3_n\), then

$$\begin{aligned} |\xi _n|^2&\le \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3|\xi _n^3|^2 = \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3\\&\qquad + \lambda _n^3P(x_n,\xi _n^3) - \lambda ^3_nP(x_n,\xi _n^3)\\&\quad = M(x_n,\xi _n) + \lambda _n^3(1-P(x_n,\xi _n^3))\le M(x_n,\xi _n) + 1 \le (C+1)|\xi _n| + 1. \end{aligned}$$

It must be

$$\begin{aligned} |\xi _n| \le (C+1) + \sqrt{2}. \end{aligned}$$

In the final possibility two points \(\xi _n^2\) and \(\xi _n^3\) are on the unit circle. Then

$$\begin{aligned} |\xi _n|^2&\le \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3|\xi _n^3|^2\\&= \lambda _n^1|\xi _n^1|^2+ \lambda _n^2+\lambda _n^2P(x_n,\xi _n^2) - \lambda ^2_nP(x_n,\xi _n^2) + \lambda _n^3 + \lambda _n^3P(x_n,\xi _n^3) - \lambda ^3_nP(x_n,\xi _n^3)\\&= M(x_n,\xi _n) + \lambda _n^2(1-P(x_n,\xi _n^2)) + \lambda _n^3(1-P(x_n,\xi _n^3))\le M(x_n,\xi _n) + 2\\&\le (C+1)|\xi _n| + 2. \end{aligned}$$

It follows that \(|\xi _n|\) is bounded, and the proof by contradiction is complete.

Remark A.5

Sometimes the assertion (N4) in the definition of an N-function is replaced with its weaker, nonuniform version

  1. (N4’)

    \( \lim _{|\xi |\rightarrow 0}\frac{M(x,\xi )}{|\xi |} = 0\qquad \text {and}\qquad \lim _{|\xi |\rightarrow \infty }\frac{M(x,\xi )}{|\xi |} = \infty \quad \text {for a.e.}\quad x\in \Omega \).

It is clear that (N4) implies (N4’). We demonstrate that if we replace (N4) with (N4’) in the definition of an N-function, then we lose the stability property. Indeed, consider \(\Omega = (0,1)\), \(d=1\) and

$$\begin{aligned} M(x,\xi ) = {\left\{ \begin{array}{ll} \frac{|\xi |^2}{x+|\xi |}\quad \text {for} \quad |\xi |\le 1,\\ \frac{|\xi |^3}{x+|\xi |}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

This function satisfies (N1)–(N3), as well as (N4’) and (N5), but it is not stable, as

$$\begin{aligned} m_2(s) = {\left\{ \begin{array}{ll} s\quad \text {for} \quad |\xi |\le 1,\\ s^2\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

which is not an N-function as it has linear growth at zero. Moreover, consider \(\Omega = (0,1)\), \(d=1\) and

$$\begin{aligned} M(x,\xi ) = {\left\{ \begin{array}{ll} \left( \frac{1}{2}+x\right) |\xi |^2\quad \text {for} \quad |\xi |\le 1,\\ |\xi |+x|\xi |^2\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

This function satisfies (N1)–(N3), (N4’), and (N5) but it does not satisfy (N4) and it is not stable, as

$$\begin{aligned} m_1(s) = {\left\{ \begin{array}{ll} \frac{1}{2}s^2\quad \text {for} \quad |\xi |\le 1,\\ s\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

and it has linear, and not superlinear growth at infinity.

Remark A.6

We will make use of the function \(M_1:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) defined as

$$\begin{aligned} M_1(\xi ) = \widetilde{\widetilde{\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } M(x,\xi )}}, \end{aligned}$$

the greatest convex minorant of \({{\,\mathrm{ess\,inf}\,}}_{x\in \Omega } M(x,\xi )\). It is not hard to verify that if M is an N-function, then \(M_1\) is a homogeneous N-function and for every \(\xi \in {\mathbb {R}}^d\) and for almost every \(x\in \Omega \) there holds

$$\begin{aligned} m_1(|\xi |) \le M_1(\xi ) \le M(x,\xi ). \end{aligned}$$

The following result shows the integrability of an N-function

Lemma A.7

Every N-function is integrable, i.e. for every \(\xi \in {\mathbb {R}}^d\) there holds

$$\begin{aligned} \int _{\Omega }M(x,\xi )\, dx < \infty . \end{aligned}$$

Proof

The result readily follows from stability of M and the fact that \(\Omega \) is bounded. \(\square \)

We continue by reminding some properties of N-functions.

Lemma A.8

If LM are N-functions, then the following assertions hold

  1. 1.

    For almost every \(x\in \Omega \) and for every \(\xi ,\eta \in {\mathbb {R}}^d\) there holds the following Fenchel–Young inequality

    $$\begin{aligned} |\xi \cdot \eta | \le M(x,\xi ) + {\widetilde{M}}(x,\eta ). \end{aligned}$$
  2. 2.

    If, for some \(x\in \Omega \), there holds \(L(x,\xi )\le M(x,\xi )\) for all \(\xi \in {\mathbb {R}}^d\) then, for this x, \({\widetilde{M}}(x,\xi )\le {\widetilde{L}}(x,\xi )\) for every \(\xi \in {\mathbb {R}}^d\).

  3. 3.

    There holds

    $$\begin{aligned} \lim _{|\xi |\rightarrow \infty } \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\xi )}{|\xi |} = \infty \quad \text {and}\quad \lim _{|\xi |\rightarrow 0}\mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\xi )}{|\xi |}=0. \end{aligned}$$
  4. 4.

    There holds

    $$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } \inf _{|\xi |=s} {\widetilde{M}}(x,\xi ) \ge \widetilde{m_2}(s) > 0\quad \text {for every nonzero}\quad \xi \in {\mathbb {R}}^d. \end{aligned}$$
  5. 5.

    There holds

    $$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega } {\widetilde{M}}(x,\xi ) \le \widetilde{m_1}(|\xi |) < \infty \quad \text {for every}\quad \xi \in {\mathbb {R}}^d. \end{aligned}$$

It follows that the complementary function of an N-function is also an N-function.

Proof

The assertions 1. and 2. of the above lemma are standard properties of the Fenchel conjugate valid for functions which satisfy (N1)–(N3) even without (N4) and (N5). To prove 3. take \(\xi = K \eta / |\eta |\) in the Fenchel–Young inequality which yields

$$\begin{aligned} K |\eta | \le {\widetilde{M}}(x,\eta ) + M(x, K\eta / |\eta |) \le {\widetilde{M}}(x,\eta ) + m_2(K). \end{aligned}$$

Dividing by \(|\eta |\) and taking \({{\,\mathrm{ess\,inf}\,}}_{x\in \Omega }\) we obtain

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\eta )}{|\eta |} \ge K - \frac{m_2(K)}{|\eta |}, \end{aligned}$$

and the assertion follows by passing with \(|\eta |\) to \(\infty \). To prove the second assertion of 3. observe that

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\xi )}{|\xi |} \le \frac{\widetilde{m_1}(|\xi |)}{|\xi |}. \end{aligned}$$

For any \(s\in (0,\infty )\) we can find \(t=t(s)\in (0,\infty )\) such that \(\widetilde{m_1}(s) + m_1(t) = st\). Such t always exists as the equality in the Fenchel–Young inequality is equivalent to the fact that \(t\in \partial m_1(s)\) (i.e. t belongs to the convex subdifferential of \(m_1\) at s), and for convex functions leading from \({\mathbb {R}}\) to \({\mathbb {R}}\) (which have to be continuous) the subdifferential is always nonempty. Let \(s_n\rightarrow 0^+\). The corresponding sequence \(t(s_n)\) must be bounded. Indeed, if this is not the case, then, for a subsequence

$$\begin{aligned} \frac{m_1(t(s_n))}{t(s_n)} \le \frac{m_1(t(s_n)) + \widetilde{m_1}(s_n)}{t(s_n)} = s_n, \end{aligned}$$

a contradiction. So, for a subsequence, \(t(s_n) \rightarrow t_0\). Passing to the limit with n to \(\infty \) in the expression \(\widetilde{m_1}(s_n) + m_1(t(s_n)) = s_n t(s_n)\) we deduce that \(t_0 = 0\) and the whole sequence converges. Now

$$\begin{aligned} \frac{\widetilde{m_1}(s_n)}{s_n} \le \frac{\widetilde{m_1}(s_n)+m_1(t(s_n))}{s_n} = t(s_n) \rightarrow 0\quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

The assertion 3. is proved. The first inequality in 4. follows from 2. The fact that \(\widetilde{m_2}(s)\ne 0\) for \(s\ne 0\) follows from the fact that \(m_2\) is an N-function. Indeed, assume the contrary, that is \( \widetilde{m_2}(s) = 0 \) for some nonzero \(s\in {\mathbb {R}}\). Taking \(t = s/K\) in the Fenchel–Young inequality we obtain

$$\begin{aligned} m_2(s/K) + \widetilde{m_2}(s) \ge s^2 / K. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{m_2(s/K)}{s/K} \ge s, \end{aligned}$$

and we have the contradiction by passing with K to infinity, as \(m_2\), an N-function, in particular satisfies the assertion (N4) of Definition A.1. Let us prove 5. To this end assume, for contradiction, that there exists \(s\in {\mathbb {R}}\) and the sequence \(\{ t_n \}\subset {\mathbb {R}}\) such that \(t_n s - m_1(t_n) \ge n\). It is clear that \(t_n \rightarrow \infty \). Then we obtain

$$\begin{aligned} s\ge \frac{m_1(t_n)}{t_n} + \frac{n}{t_n} \ge \frac{m_1(t_n)}{t_n}, \end{aligned}$$

and we get the contradiction with superlinear growth of \(m_1\) at infinity by passing with n to infinity. \(\square \)

Sometimes we will use the so called \(\Delta _2\) condition which states that there exists a constant \(c>0\) and a nonnegative function \(h\in L^1(\Omega )\) such that for a.e. \(x\in \Omega \) and every \(\xi \in {\mathbb {R}}^d\)

$$\begin{aligned} M(x,2\xi ) \le c M(x,\xi ) + h(x). \end{aligned}$$
(A.2)

If M satisfies the above condition for \(\xi \in {\mathbb {R}}^d \setminus B(0,R)\) for some \(R>0\) than we write that \(M \in \Delta _2^\infty \) and we say that M satisfies \(\Delta _2\) far from origin.

Orlicz–Musielak spaces. We remind the definition of the Orlicz–Musielak class \({\mathcal {L}}_M(\Omega )\) and two spaces \(L_M(\Omega )\), and \(E_M(\Omega )\).

Definition A.9

Suppose that M is an N-function.

  1. 1.

    \({\mathcal {L}}_M(\Omega )\), the Orlicz–Musielak class, is the set of all measurable functions \(\xi :\Omega \rightarrow {\mathbb {R}}^d\) such that

    $$\begin{aligned} \int _{\Omega } M(x,\xi (x))\, dx < \infty . \end{aligned}$$
  2. 2.

    \(L_M(\Omega )\) is the generalized Orlicz–Musielak space, which is the smallest linear space containing \({\mathcal {L}}_M(\Omega )\), equipped with the Luxemburg norm

    $$\begin{aligned} \Vert \xi \Vert _{L_M} = \inf \left\{ \lambda > 0\, :\ \int _{\Omega } M\left( x,\frac{\xi (x)}{\lambda }\right) \, dx \le 1 \right\} . \end{aligned}$$
  3. 3.

    \(E_M(\Omega )\) is the closure of \(L^\infty (\Omega )^d\) in \(L_M\) norm.

We prove the following result

Theorem A.10

The space \(C^\infty _0(\Omega )^d\) is dense in \(E_M(\Omega )\) in \(L_M\) norm.

Proof

Since \(E_M(\Omega )\) is a closure of \(L^\infty (\Omega )^d\) in \(L_M\) norm, it is enough to prove that \(C^\infty _0(\Omega )^d\) is dense in \(L^\infty (\Omega )^d\) in \(L_M(\Omega )\) norm. We first prove that \(C_c(\Omega )^d\) is \(L_M\) dense in \(L^\infty (\Omega )^d\). Take \(v\in L^\infty (\Omega )^d\). By the Luzin theorem there exists a sequence of compact sets \(E_n\) such that \(|\Omega \setminus E_n| < 1/n\) and the functions \(v|_{E_n}:E_n\rightarrow {\mathbb {R}}\) are continuous. These functions, by the Tietze–Urysohn lemma can be extended to functions \(v_n\) in \(C_c(\Omega )^d\) such that \(\Vert v_n\Vert _{L^\infty } \le \Vert v\Vert _{L^\infty }\). Now

$$\begin{aligned}&\int _{\Omega }M\left( x,\frac{v_n(x)-v(x)}{\lambda }\right) \, dx = \int _{\Omega \setminus {E_n}}M\left( x,\frac{v_n(x)-v(x)}{\lambda }\right) \, dx \\&\quad \le \int _{\Omega \setminus {E_n}} m_2 \left( \frac{|v_n(x)|+|v(x)|}{\lambda }\right) \,dx\\&\quad \le m_2 \left( \frac{2\Vert v\Vert _{L^\infty (\Omega )^d}}{\lambda }\right) |\Omega \setminus {E_n}| \rightarrow 0 \quad \text {as}\quad n\rightarrow \infty \quad \text {for every}\ \ \lambda >0. \end{aligned}$$

We now prove that \(C^\infty _0(\Omega )^d\) is \(L_M\) dense in \(C_c(\Omega )^d\). To this end take \(v\in C_c(\Omega )^d\) and extend it to \({\mathbb {R}}^d\) by taking \(v=0\) outside \(\Omega \). Choose a standard mollifier kernel

$$\begin{aligned} \theta _\epsilon (x) = \frac{1}{\epsilon ^n}\theta \left( \frac{x}{\epsilon }\right) \quad \text {where}\quad \theta (x) = {\left\{ \begin{array}{ll} Ce^{\frac{1}{|x|^2}-1}\quad \text {for}\quad |x|\le 1,\\ 0\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

where the constant C is chosen such that \(\int _{{\mathbb {R}}^n}\theta (x)\, dx = 1\). Define \(v^\epsilon (x) = \int _{{\mathbb {R}}^d} \theta _\epsilon (x-y)v(y)\, dy\). Then, if only \(\epsilon \) is small enough, \(v^\epsilon \in C^\infty _0({\mathbb {R}}^d)\) and there holds the pointwise convergence \(\lim _{\epsilon \rightarrow 0}v^\epsilon (x) = v(x)\) for almost every \(x\in \Omega \). Moreover, \(|v^\epsilon (x)| \le \Vert v\Vert _{L^\infty (\Omega )^d}\) for almost every \(x\in \Omega \). There holds

$$\begin{aligned} M\left( x,\frac{v_\epsilon (x)-v(x)}{\lambda }\right) \le m_2 \left( \frac{|v_\epsilon (x)|+|v(x)|}{\lambda }\right) \le m_2 \left( \frac{2\Vert v\Vert _{L^\infty (\Omega )^d}}{\lambda }\right) \quad \text {for every}\ \ \lambda >0. \end{aligned}$$

We can use the Lebesgue dominated convergence theorem to deduce that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\int _{\Omega } M\left( x,\frac{v_\epsilon (x)-v(x)}{\lambda }\right) \, dx = 0\quad \text {for every}\ \ \lambda >0, \end{aligned}$$

whence \(\lim _{\epsilon \rightarrow 0}\Vert v_\epsilon -v\Vert _{L_M} = 0\), and the proof is complete. \(\square \)

It is not hard to verify, that for \(v\in L_M(\Omega )\) there holds

$$\begin{aligned}&\Vert v\Vert _{L_M} \le \int _{\Omega }M(x,v(x))\, dx+1, \end{aligned}$$
(A.3)
$$\begin{aligned}&\Vert v\Vert _{L_M} \le 1 \Rightarrow \int _{\Omega }M(x,v(x))\, dx \le \Vert v\Vert _{L_M}. \end{aligned}$$
(A.4)

It is clear that \(L^\infty (\Omega )^d \subset E_M(\Omega )\) and \({\mathcal {L}}_M(\Omega ) \subset L_M(\Omega )\). We also observe that \(E_M(\Omega )\subset {\mathcal {L}}_M(\Omega )\). Indeed, if \(v_k\) is a sequence in \(L^\infty (\Omega )^d\) such that \(\Vert v_k-v\Vert _{L_M} \rightarrow 0\), then

$$\begin{aligned} \int _{\Omega }M(x,v(x))\, dx \le \frac{1}{2}\left( \int _{\Omega }M(x,2v_k(x))\, dx+\int _{\Omega }M(x,2(v(x)-v_k(x)))\, dx\right) . \end{aligned}$$

Taking k large enough, such that \(2\Vert v-v_k\Vert _{L_M} \le 1\), it follows that

$$\begin{aligned} \int _{\Omega }M(x,v(x))\, dx&\le \frac{1}{2}\int _{\Omega }M(x,2v_k(x))\, dx+\Vert v-v_k\Vert _{L_M} \\&\quad \le \frac{1}{2}|\Omega | m_2(2\Vert v_k\Vert _{L^\infty }))+\Vert v-v_k\Vert _{L_M} < \infty . \end{aligned}$$

Some functional analytic properties of the defined spaces are summarized in the following lemmas [16, 41, 57, 58, 64].

Lemma A.11

If M is an N-function, \(\xi \in L_M(\Omega )\), and \(\eta \in L_{{\widetilde{M}}}(\Omega )\), then the following generalized Hölder inequality holds

$$\begin{aligned} \left| \int _{\Omega } \xi \cdot \eta \, dx\right| \le 2 \Vert \xi \Vert _{L_M(\Omega )}\Vert \eta \Vert _{L_{{\widetilde{M}}}(\Omega )} \end{aligned}$$

Lemma A.12

Let M be an N-function. Then

  • \(E_M(\Omega )\) is separable.

  • \(E_M(\Omega )^* = L_{{\widetilde{M}}}(\Omega )\).

  • \(E_M(\Omega ) = L_M(\Omega )\) if and only if \(M\in \Delta _2^\infty \).

  • \(L_M(\Omega )\) is separable if and only if \(M\in \Delta _2^\infty \).

  • \(L_M(\Omega )\) is reflexive if and only if both \(M\in \Delta _2^\infty \) and \({\widetilde{M}}\in \Delta _2^\infty \).

In this article we assume nowhere that both M and \({\widetilde{M}}\) satisfy the \(\Delta _2\) condition, so that we have to deal with the lack of reflexivity. We also deal with the case where M does not satisfy the \(\Delta _2\) condition, so we cannot use the separability of \(L_M\). Despite this difficulties we are still in position to obtain the existence results using the functional analytic tools developed in [16, 40, 44, 45].

If M is an N-function, we define the space

$$\begin{aligned} V_0^M = \{ v\in W^{1,1}_0(\Omega )\, :\ \nabla v \in L_M(\Omega ) \}. \end{aligned}$$

We will need the following version of the modular Poincaré inequality, cf. [13, Lemma 1], [61, Lemma 3], or more recent works [40, Theorem 2.2], [26, Corollary 4.2].

Theorem A.13

Let \(m:[0,\infty )\rightarrow [0,\infty )\) be an N-function. There exist constants \(\lambda >0\) and \(C>0\) such that for every \(u\in W^{1,1}_0(\Omega )\) satisfying \(\int _{\Omega }m(\lambda |\nabla u|)\, dx < \infty \) there holds

$$\begin{aligned} \int _{\Omega }m(|u|)\, dx \le C \int _{\Omega }m(\lambda |\nabla u|)\, dx. \end{aligned}$$

Remark A.14

In [40] it is proved that the above theorem holds with \(\lambda =1\) provided m satisfies the \(\Delta _2\) condition. A careful analysis of its proof, however, reveals that without the \(\Delta _2\) condition the result holds with a constant \(\lambda \) not necessary equal to one, but dependent only on \(\Omega \) and d.

We remind the definition of modular convergence, cf. [16, 40, 41, 43].

Definition A.15

A sequence \(\{ v_m \}_{m=1}^\infty \) of measurable \({\mathbb {R}}^d\) valued function on \(\Omega \) is said to converge modularly to a function v if there exists \(\lambda >0\) such that

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _{\Omega } M\left( x,\frac{v_m-v}{\lambda }\right) \, \, dx = 0. \end{aligned}$$

We denote the modular convergence by \(v_m \xrightarrow {M} v\). Equivalently, cf. [41, Lemma 2.1], \(\{v_m\}_{m=1}^\infty \) converges modularly to v if \(v_m\rightarrow v\) in measure and the sequence \(\{ M(\cdot , \lambda v_m) \}_{m=1}^\infty \) is uniformly integrable for some \(\lambda >0\).

Lemma A.16

(cf. [41, Lemma 2.2]) Let M be an N-function. If, for constants \(c,\lambda >0\), we have \(\int _{\Omega }M(x,\lambda v_m)\, dx \le c\) for all \(m\in {\mathbb {N}}\) then the sequence \(\{v_m\}_{m=1}^\infty \) is uniformly integrable.

Proof

For every \(m\in {\mathbb {N}}\) it holds

$$\begin{aligned} c\ge \int _{\{x\in \Omega \, :\ |v_m(x)|\ge R\}} M(x,\lambda v_m(x))\, dx \ge \lambda \int _{\{x\in \Omega \, :\ |v_m(x)|\ge R\}} \frac{m_1(\lambda |v_m(x)|)}{\lambda |v_m(x)|} |v_m(x)|\, dx. \end{aligned}$$

As \(m_1\) is an N-function, for any \(D>0\) there exists \(R_0(D)>0\) such that for any \(s\ge R_0\) there holds \(\frac{m_1(\lambda s)}{\lambda s} \ge D\). Hence

$$\begin{aligned} \frac{c}{\lambda D} \ge \int _{\{x\in \Omega \, :\ |v_m(x)|\ge R_0(D)\}} |v_m(x)|\, dx, \end{aligned}$$

and the assertion follows easily. \(\square \)

The following approximation theorem which has been proved in [27, Theorem 3.1] is valid in nonreflexive and nonseparable Musielak–Orlicz spaces.

Theorem A.17

Let \(\Omega \) be a Lipschitz domain and let an N-function M satisfy (C2). Then for any \(u\in L^{\infty }(\Omega )\cap V_0^M\) there exists a sequence \(\{u_m\}_{m=1}^\infty \) of functions belonging to \(C_0^\infty (\Omega )\) such that \(u_m\rightarrow u\) strongly in \(L^1(\Omega )\) and \(\nabla u_m \xrightarrow {M} \nabla u\) in \(L_M(\Omega ).\) Moreover, there exists a constant \(c=c(\Omega ) > 0\), such that \(\Vert u_m\Vert _{L^{\infty }(\Omega )}\le c\Vert u\Vert _{L^{\infty }(\Omega )}.\)

We remind an important property of the modular convergence

Lemma A.18

(cf. [41, Proposition 2.2]) Suppose that the sequences \(\{v_k\}_{k=1}^\infty \) and \(\{w_k\}_{k=1}^\infty \) are uniformly bounded in \(L_M(\Omega )\) and \(L_{{\widetilde{M}}}(\Omega )\), respectively. If \(v_k\xrightarrow {M} v\) and \(w_k\xrightarrow {{\widetilde{M}}} w\) then \(v_k\cdot w_k \rightarrow v\cdot w\) in \(L^1(\Omega )\).

Appendix B: Some useful tools of nonlinear analysis

We recall some tools useful in the arguments of this article.

Definition B.1

A sequence \(\{f_m\}_{m=1}^\infty \) of measurable functions \(f_m:\Omega \rightarrow {\mathbb {R}}\) is uniformly integrable if, equivalently, one of the following conditions holds:

  1. (i)
    $$\begin{aligned} \lim _{R\rightarrow \infty }\left( \sup _{m\in {\mathbb {N}}}\int _{\{ x\in \Omega \, :\ |f_m(x)| \ge R \}}|f_m(x)|\, dx\right) = 0. \end{aligned}$$
  2. (ii)
    $$\begin{aligned} \forall \, \epsilon>0\ \ \exists \, \delta >0\quad \sup _{m\in {\mathbb {N}}} \int _{\Omega }\left( |f_m(x)|-\frac{1}{\sqrt{\delta }}\right) _+\, dx \le \epsilon . \end{aligned}$$
  3. (iii)

    There exists a continuous, concave, and nondecreasing function \(\omega :[0,\infty ) \rightarrow [0,\infty )\) such that \(\omega (0)=0\) and for every measurable set \(E\subset \Omega \) and for every \(m\in {\mathbb {N}}\) there holds

    $$\begin{aligned} \int _{E}|f_m(x)|\, dx \le \omega (|E|). \end{aligned}$$
  4. (iv)

    There exists a function \(\Phi :[0,\infty )\rightarrow [0,\infty )\) which is convex, \(\Phi (0) = 0\), \(\lim _{s\rightarrow \infty }\frac{\Phi (s)}{s} = \infty \), and

    $$\begin{aligned} \sup _{n\in {\mathbb {N}}} \int _{\Omega }\Phi (|f_n(x)|)\, dx < \infty . \end{aligned}$$
  5. (v)

    The set \(\{ f_n\}_{n=1}^\infty \) is relatively compact (or, equivalently, relatively sequentially compact) in the weak topology of \(L^1(\Omega )\).

Remark B.2

The fact that condition (iv) of the above definition is equivalent to the other three is known as the de la Vallée Poussin theorem. The equivalence of assertion (v) to the remaining ones is known as the Dunford–Pettis theorem.

Young measures are now a standard tool of nonlinear analysis, we refer for example to [55] for a comprehensive exposition of their theory. We will need the following version of the generalized fundamental theorem on Young measures from [47], where by \({\mathcal {M}}({\mathbb {R}}^N)\) we denote the space of bounded Radon measures.

Proposition B.3

(cf. [47, Theorem 4.1]) Let \(\Omega \) be an open and bounded subset of \({\mathbb {R}}^d.\) Assume that the sequence \(\{ \nu ^j \}_{j=1}^\infty \subset L_{w}^{\infty }(\Omega , {\mathcal {M}}({\mathbb {R}}^N))\) of weakly-* measurable mappings is such that \(\nu ^j(x)=\nu ^j_x\) is a probability measure for almost every \(x\in \Omega \). If the sequence \(\nu ^j\) satisfies the tightness condition:

$$\begin{aligned} \lim _{M\rightarrow \infty } \sup _j |\{x\in \Omega \,\ \mathrm {supp}(\nu ^j_x)\setminus B(0,M)\ne \emptyset \}| \rightarrow 0, \end{aligned}$$

then the following assertions are true.

  1. (1)

    There exists a weakly-* measurable mapping \(\nu \in L^\infty _w(\Omega ,{\mathcal {M}}({\mathbb {R}}^N))\) such that, for a subsequence still denoted by j, there holds

    $$\begin{aligned} \nu ^j\rightarrow \nu \quad \mathrm {weakly}-*\ \mathrm {in} \quad L_w^{\infty }(\Omega , {\mathcal {M}}({\mathbb {R}}^N)), \end{aligned}$$
  2. (2)

    \(\Vert \nu _x\Vert _{{\mathcal {M}}({\mathbb {R}}^N)}=1\) a.e in \(\Omega .\) Moreover, for every \(f\in L^{\infty }(\Omega ,C_b({\mathbb {R}}^N))\) there holds

    $$\begin{aligned} \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x^j(\lambda )\rightarrow \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x(\lambda ) ~~\mathrm {weakly}-*\ \mathrm {in}~~L^\infty (\Omega ). \end{aligned}$$
  3. (3)

    For every measurable subset \(A\subset \Omega \) and for every Carathéodory function f (measurable in the first, and continuous in the second variable) such that

    $$\begin{aligned} \lim _{R\rightarrow 0}\sup _{j\in {\mathbb {N}}}\int _A\int _{\{\lambda \in {\mathbb {R}}^N\, :\ |f(x,\lambda )| > R\}} |f(x,\lambda )|d\nu ^j_x(\lambda )dx=0, \end{aligned}$$

    there holds

    $$\begin{aligned} \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x^j(\lambda )\rightarrow \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x(\lambda ) ~~\mathrm {weakly}\ \ \mathrm {in}~~L^1(A) \end{aligned}$$

The following corollary is the generalization of the result on the lower-semicontinuity of Young measure generated by a sequence of functions, cf. [54, Corollary 3.3], to the case when the measure is generated by a sequence of measures.

Corollary B.4

Let \( \Omega \subset {\mathbb {R}}^d\) be an open and bounded subset of \({\mathbb {R}}^d\). Suppose that

$$\begin{aligned} \{ \nu ^j \}_{j=1}^\infty \subset L_{w}^{\infty }(\Omega , {\mathcal {M}}({\mathbb {R}}^N))\quad \mathrm {and}\quad \nu \in L^\infty _w(\Omega , {\mathcal {M}}({\mathbb {R}}^N)) \end{aligned}$$

are weakly-* measurable mappings such that \(\nu ^j_x\) and \(\nu _x\) are probability measures for a.e. \(x\in \Omega \). Moreover assume that

$$\begin{aligned} \nu ^{j} \rightarrow \nu \quad \mathrm {weakly}-*\ \ \mathrm {in}\quad L^\infty _w(\Omega ,{\mathcal {M}}({\mathbb {R}}^N)). \end{aligned}$$

Then for any measurable set \( E\subset \Omega \) and every Carathéodory function such that there exists \(m\in L^1(\Omega ), \) with \(m\ge 0\) and \(f(x,\lambda )> - m(x)\) for almost every \(x\in \Omega \) and every \(\lambda \in \bigcup _{j=1}^\infty \mathrm {supp}\, \nu _x^j\) , there holds

$$\begin{aligned} \int _{E}\int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x(\lambda )\le \liminf _{j\rightarrow \infty } \int _{E}\int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x^j(\lambda ) \end{aligned}$$

Proof

The proof follows the lines of the proof of Corollary 3.3 in [54]. First assume that there exist \(R>0\) such that \(f(x,\lambda )=0\) for \( |\lambda |\ge R\). By the Scorza–Dragoni theorem there exists an increasing sequence of compact sets \(E_k\) such that \(|E \setminus E_k| \rightarrow 0\) as \(k\rightarrow \infty \) and \(f|_{E_k\times \mathbb {R^N}}\) is continuous. Define \(F_k: E\rightarrow C_0({\mathbb {R}}^N)\) as \(F_k(x)= \chi _{E_k}(x)f(x,\cdot ).\) We observe that \(F_k \in L^1(E,C_0({\mathbb {R}}^N))\). Indeed,

$$\begin{aligned}&\int _{E}\Vert F_k(x)\Vert _{C_0({\mathbb {R}}^N)}dx=\int _{E}\sup _{\lambda \in {\mathbb {R}}^N}|F_k(x,\lambda )|dx=\int _{E_k}\sup _{\lambda \in {\mathbb {R}}^N}|f(x,\lambda )|dx \\&\qquad \le |\Omega | \sup _{(x,\lambda )\in E_k\times {\mathbb {R}}^N}|f(x,\lambda )|= |\Omega | \sup _{(x,\lambda )\in E_k\times \overline{B(-R,R)}}|f(x,\lambda )| < \infty . \end{aligned}$$

Now, as \((L^1(E,C_0({\mathbb {R}}^N)))' = L^\infty _w(\Omega ,{\mathcal {M}}({\mathbb {R}}^N))\), there holds

$$\begin{aligned}&\lim _{j\rightarrow \infty }\int _{E}\int _{{\mathbb {R}}^N}F_k(x,\lambda )d\nu _x^{j}(\lambda )dx= \int _{E}\int _{\mathbb {R^N}}F_k(x,\lambda )d\nu _x(\lambda )dx, \\&\lim _{j\rightarrow \infty } \int _{E_k}\int _{{\mathbb {R}}^N}(f(x,\lambda )+m(x)-m(x))d\nu _x^{j}(\lambda )dx\\&\quad =\int _{E_k}\int _{\mathbb {R^N}}(f(x,\lambda )+m(x)-m(x))d\nu _x(\lambda )dx. \end{aligned}$$

As \(\nu ^j_x\) and \(\nu \) are probability measures, it follows that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{E_k}\int _{{\mathbb {R}}^N}(f(x,\lambda )+m(x))d\nu _x^{j}(\lambda )dx=\int _{E_k}\int _{\mathbb {R^N}}(f(x,\lambda )+m(x))d\nu _x(\lambda )dx. \end{aligned}$$

It follows that

$$\begin{aligned} \int _{E}\int _{\mathbb {R^N}}\chi _{E_k}(x)(f(x,\lambda )+m(x))d\nu _x(\lambda )dx \le \liminf _{j\rightarrow \infty } \int _{E}\int _{{\mathbb {R}}^N}(f(x,\lambda )+m(x))d\nu _x^{j}(\lambda )dx. \end{aligned}$$

Letting \(k\rightarrow \infty \) we obtain the assertion by the monotone convergence theorem. To remove the assumption that \(f(x,\lambda ) = 0\) \(|\lambda |\ge R\) consider an increasing sequence of nonnegative functions \(\eta _l\subset C_0^{\infty }({\mathbb {R}}^N)\) that converges pointwise to 1. We use the above result for \(f(x,\lambda )\eta _l(\lambda )\)

$$\begin{aligned}&\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )\eta _l(\lambda ) d\nu _x(\lambda )dx\le \liminf _{j\rightarrow \infty }\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )\eta _l(\lambda )d\nu _x^{j}(\lambda )dx\\&\qquad \le \liminf _{j\rightarrow \infty }\left( \int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d \nu _x^{\epsilon }(\lambda )dx - \int _{E}\int _{\mathbb {R^N}}m(x)\eta ^l(\lambda ) d\nu _x^{j}(\lambda )dx\right) . \end{aligned}$$

But \(m(x)\eta ^l(\lambda ) \in L^1(E,C_0({\mathbb {R}}^N))\) and hence

$$\begin{aligned} \int _{E}\int _{\mathbb {R^N}}(f(x,\lambda )+m(x))\eta _l(\lambda ) d\nu _x(\lambda )dx \le \liminf _{j\rightarrow \infty }\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d \nu _x^{j}(\lambda )dx. \end{aligned}$$

We can pass to the limit \(l\rightarrow \infty \) in the left-hand side by the monotone convergence theorem

$$\begin{aligned} \int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d\nu _x(\lambda )dx \le \liminf _{j\rightarrow \infty }\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d \nu _x^{j}(\lambda )dx, \end{aligned}$$

and the assertion follows. \(\square \)

We recall the definition of the biting convergence and the statement of the Chacon biting Lemma.

Definition B.5

Let \(\Omega \subset {\mathbb {R}}^d\) be a measurable set. We say that a sequence \(\{ f_j \}_{j=1}^\infty \subset L^1(\Omega )\) converges to a function \(f\in L^1(\Omega )\) in a biting sense (and we write \(f_j {\mathop {\rightarrow }\limits ^{b}} f\)) if there exists a sequence of measurable sets \(E_l \subset \Omega \) with \(|\Omega \setminus E_l| \rightarrow 0\) as \(l\rightarrow \infty \) and \(E_1\subseteq E_2 \subseteq \cdots \subseteq E_l \subseteq \cdots \subseteq \Omega \) such that

$$\begin{aligned} f_j \rightarrow f\quad \mathrm {weakly}\ \mathrm {in}\ L^1(E_l)\quad \mathrm {for}\ \mathrm {every} \ l\in {\mathbb {N}}. \end{aligned}$$

The proof of the following proposition (known as the Chacon biting lemma) can be found for example in [7].

Proposition B.6

Let \(\Omega \subset {\mathbb {R}}^d\) be a measurable set and let the sequence \(\{ f_j \}_{j=1}^\infty \subset L^1(\Omega )\) be bounded in \(L^1(\Omega )\). There exists a subsequence of indices, still denoted by j, and a function \(f\in L^1(\Omega )\) such that \(f_j {\mathop {\rightarrow }\limits ^{b}} f\).

We will need the following result which states when the sequence which converges in the biting sense is convergent weakly in \(L^1(\Omega )\), cf. [46, Lemma 4.6].

Proposition B.7

Let the sequence \(\{ a_j \}_{j=1}^\infty \subset L^1(\Omega )\) and let \(0\le a_0\in L^1(\Omega )\) be such that \(a_j(x) \ge -a_0(x)\) for almost every \(x\in \Omega \). If

$$\begin{aligned} a_n {\mathop {\rightarrow }\limits ^{b}} a \quad \mathrm {and} \quad \limsup _{j\rightarrow \infty }\int _{\Omega }a_j\, dx \le \int _{\Omega } a\, dx, \end{aligned}$$

then

$$\begin{aligned} a_j \rightarrow a\quad \mathrm {weakly} \ \mathrm {in}\quad L^1(\Omega ). \end{aligned}$$

We also make use of the following well known result

Proposition B.8

Assume that \(\Omega \subset {\mathbb {R}}^d\) is a bounded set. Let the sequence \(f_j\rightarrow f\) weakly in \(L^1(\Omega )\), and let \(g_j, g\in L^\infty (\Omega )\) be such that \(\Vert g_j\Vert _{L^\infty (\Omega )} \le C\) and \(\Vert g\Vert _{L^\infty (\Omega )} \le C\), where the constant C is independent of j and for almost every \(x\in \Omega \) there holds the pointwise convergence \(g_j(x)\rightarrow g(x)\). Then

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{\Omega }f_j g_j\, dx = \int _{\Omega }f g\, dx \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Denkowska, A., Gwiazda, P. & Kalita, P. On renormalized solutions to elliptic inclusions with nonstandard growth. Calc. Var. 60, 21 (2021). https://doi.org/10.1007/s00526-020-01893-4

Download citation

Mathematics Subject Classification

  • 35J60
  • 35D30