Entropy and renormalized solutions for a nonlinear elliptic problem in Musielak--Orlicz spaces

L. M. Kozhevnikova

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 1, pp. 98-115.

Résumé

In this paper, we establish the equivalence of entropy and renormalized solutions of second-order elliptic equations with nonlinearities defined by the Musielak–Orlicz functions and the right-hand side from the space $L_1(\Omega).$ In nonreflexive Musielak—Orlicz—Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary.

Détail

Keywords: second-order elliptic equation, entropy solution, renormalized solution, Musielak–Orlicz–Sobolev space, existence and uniqueness of solutions.

Bibliographie

[1] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[2] Kozhevnikova L. M., “Entropiinye i renormalizovannye resheniya anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei”, Mat. sb., 210:3 (2019), 131–161 | DOI | MR | Zbl

[3] Kovalevskii A. A., Skrypnik I. I., Shishkov A. E., Singulyarnye resheniya nelineinykh ellipticheskikh i parabolicheskikh uravnenii, Naukova dumka, Kiev, 1962 | MR

[4] Kozhevnikova L. M., Kashnikova A. P., “Ekvivalentnost entropiinykh i renormalizovannykh reshenii nelineinoi ellipticheskoi zadachi v prostranstvakh Muzilaka—Orlicha”, Diff. uravn., 59 (2023), 35–51

[5] Rutitskii Ya. B., Krasnoselskii M. A., Vypuklye funktsii i prostranstva Orlicha, Fizmatlit, M., 1958 | MR

[6] Ahmida Y., Chlebicka I., Gwiazda P., Youssfi A., “Gossez's approximation theorems in Musielak—Orlicz—Sobolev spaces”, J. Funct. Anal., 275:9 (2018), 2538–2571 | DOI | MR | Zbl

[7] Ait Khellou M., Benkirane A., “Renormalized solution for nonlinear elliptic problems with lower order terms and $L^1$ data in Musielak—Orlicz spaces”, An. Univ. Craiova Ser. Mat. Inform., 43:2 (2016), 164–187 | MR | Zbl

[8] Ait Khellou M., Douiri S. M., El Hadfi Y., “Existence of solutions for some nonlinear elliptic equations in Musielak spaces with only the Log-Hölder continuity condition”, Mediterr. J. Math., 17:1 (2020), 1–18 | DOI | MR | Zbl

[9] Benilan Ph., Boccardo L., Gallouët Th., Gariepy R., Pierre M., Vazquez J. L., “An $L^1$-theory of existence and nuniqueness of solutions of nonlinear elliptic equations”, Ann. Sc. Norm. Super. Pisa Cl. Sci, 22:2 (1995), 241–273 | MR | Zbl

[10] Benkirane A., Sidi El Vally M., “An existence result for nonlinear elliptic equations in Musielak—Orlicz—Sobolev spaces”, Bull. Belg. Math. Soc. Simon Stevin, 20:1 (2013), 57–75 | DOI | MR | Zbl

[11] Benkirane A., Sidi El Vally M., “Variational inequalities in Musielak—Orlicz—Sobolev spaces”, Bull. Belg. Math. Soc. Simon Stevin, 21:5 (2014), 787–811 | DOI | MR | Zbl

[12] Boccardo L., Gallouët Th., “Nonlinear elliptic equations with right-hand side measures”, Commun. Part. Differ. Equ, 17:3-4 (1992), 641–655 | DOI | MR | Zbl

[13] Chlebicka I., “A pocket guide to nonlinear differential equations in Musielak—Orlicz spaces”, Nonlinear Anal., 175 (2018), 1–27 | DOI | MR | Zbl

[14] Chlebicka I., “Measure data elliptic problems with generalized Orlicz growth”, Proc. Roy. Soc. Edinburgh Sect. A, 153:2 (2023), 588–618 | DOI | MR

[15] Denkowska A., Gwiazda P., Kalita P., “On renormalized solutions to elliptic inclusions with nonstandard growth”, Calc. Var. Partial Differ. Equ, 60:21 (2021), 1–44 | MR

[16] Elarabi R., Rhoudaf M., Sabiki H., “Entropy solution for a nonlinear elliptic problem with lower order term in Musielak—Orlicz spaces”, Ric. Mat., 67:2 (2018), 549–579 | DOI | MR | Zbl

[17] Elemine Vall M. S. B., Ahmedatt T., Touzani A., Benkirane A., “Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with $L^1$ data”, Bol. Soc. Parana Mat., 36:1 (2018), 125–150 | DOI | MR | Zbl

[18] Gwiazda P., Skrzypczaka I., Zatorska-Goldstein A., “Existence of renormalized solutions to elliptic equation in Musielak—Orlicz space”, Differ. Equ., 264 (2018), 341–377 | DOI | MR | Zbl

[19] Gwiazda P., Świerczewska-Gwiazda A., Wróblewska A., “Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids”, Math. Methods Appl. Sci., 2010, no. 2, 125–137 | DOI | MR | Zbl

[20] Musielak J., Orlicz spaces and modular spaces, Springer, Berlin, 1983 | MR | Zbl

[21] Talha A., Benkirane A., “Strongly nonlinear elliptic boundary value problems in Musielak—Orlicz spaces”, Monatsh. Math., 186:4 (2018), 745–776 | DOI | MR | Zbl

[22] Ying Li, Fengping Y., Shulin Zh., “Entropy and renormalized solutions to the general nonlinear elliptic equations in Musielak—Orlicz spaces”, Nonlinear Anal. Real World Appl., 61 (2021), 1–20 | MR