Equivalence of solutions for non-homogeneous $ p(x) $-Laplace equations

María Medina; Pablo Ochoa; María Medina; Pablo Ochoa

doi:10.3934/mine.2023044

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-19. doi: 10.3934/mine.2023044

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Equivalence of solutions for non-homogeneous $ p(x) $-Laplace equations

María Medina ¹,
Pablo Ochoa ^{2
,
,}

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
2.
Facultad de Ingeniería, Universidad Nacional de Cuyo. CONICET. Parque Gral. San Martín 5500. Mendoza, Argentina

Received: 06 December 2021 Revised: 23 June 2022 Accepted: 23 June 2022 Published: 12 July 2022

We establish the equivalence between weak and viscosity solutions for non-homogeneous $ p(x) $-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the $ p(x) $-Laplacian compared to the constant case are the presence of $ \log $-terms and the lack of the invariance under translations.
- nonlinear elliptic equations,
- p(x)-Laplacian,
- viscosity solutions,
- weak solutions,
- comparison principle
Citation: María Medina, Pablo Ochoa. Equivalence of solutions for non-homogeneous $ p(x) $-Laplace equations[J]. Mathematics in Engineering, 2023, 5(2): 1-19. doi: 10.3934/mine.2023044

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Abstract

We establish the equivalence between weak and viscosity solutions for non-homogeneous $ p(x) $-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the $ p(x) $-Laplacian compared to the constant case are the presence of $ \log $-terms and the lack of the invariance under translations.

References

[1]	B. Barrios, M. Medina, Equivalence of weak and viscosity solutions in fractional non-homogeneous problems, Math. Ann., 381 (2021), 1979–2012. https://doi.org/10.1007/s00208-020-02119-w doi: 10.1007/s00208-020-02119-w
[2]	J. E. M. Braga, R. A. Leitao, J. E. L. Oliveira, Free boundary theory for singular/degenerate nonlinear equations with right hand side: a non-variational approach, Calc. Var., 59 (2020), 86. https://doi.org/10.1007/s00526-020-01733-5 doi: 10.1007/s00526-020-01733-5
[3]	Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383–1406. https://doi.org/10.1137/050624522 doi: 10.1137/050624522
[4]	M. G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1–67. https://doi.org/10.1090/S0273-0979-1992-00266-5 doi: 10.1090/S0273-0979-1992-00266-5
[5]	l. Diening, P. Harjulehto, P. Hästö, M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8
[6]	D. Edmunds, J. Rákosník, Sobolev embeddings with variable exponent, Stud. Math., 143 (2000), 267–293. https://doi.org/10.4064/sm-143-3-267-293 doi: 10.4064/sm-143-3-267-293
[7]	X. Fan, D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. Theor., 36 (1999), 295–318. https://doi.org/10.1016/S0362-546X(97)00628-7 doi: 10.1016/S0362-546X(97)00628-7
[8]	X. Fan, D. Zhao, The quasi-minimizer of integral functionals with $m(x)$-growth conditions, Nonlinear Anal. Theor., 39 (2000), 807–816. https://doi.org/10.1016/S0362-546X(98)00239-9 doi: 10.1016/S0362-546X(98)00239-9
[9]	X. Fan, Y. Zhao, Q. Zhang, A strong maximum principle for $p(x)$-Laplace equations, Chinese Journal of Contemporary Mathematics, 21 (2000), 277–282.
[10]	X. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{1, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424–446. https://doi.org/10.1006/jmaa.2000.7617 doi: 10.1006/jmaa.2000.7617
[11]	X. Fan, Q. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. Theor., 52 (2003), 1843–1852. https://doi.org/10.1016/S0362-546X(02)00150-5 doi: 10.1016/S0362-546X(02)00150-5
[12]	F. Ferrari, C. Lederman, Regularity of flat free boundaries for a $p(x)$-Laplacian problem with right hand side, Nonlinear Anal., 212 (2021), 112444. https://doi.org/10.1016/j.na.2021.112444 doi: 10.1016/j.na.2021.112444
[13]	H. Ishii, On the equivalence of two notions of solutions, viscosity solutions and distribution solutions, Funkcislaj Ekvacioj, 38 (1995), 101–120.
[14]	V. Julin, P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p-$Laplace equation, Commun. Part. Diff. Eq., 37 (2012), 934–946. https://doi.org/10.1080/03605302.2011.615878 doi: 10.1080/03605302.2011.615878
[15]	P. Juutinen, P. Lindqvist, A theorem of Radó type for solutions of a quasi-linear equation, Math. Res. Lett., 11 (2004), 31–34. https://doi.org/10.4310/MRL.2004.v11.n1.a4 doi: 10.4310/MRL.2004.v11.n1.a4
[16]	P. Juutinen, P. Lindqvist, J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM J. Math. Anal., 33 (2001), 699–717. https://doi.org/10.1137/S0036141000372179 doi: 10.1137/S0036141000372179
[17]	P. Juutinen, T. Lukkari, M. Parviainen, Equivalence of viscosity solutions and weak solutions for the $p(x)$-Laplacian, Ann. Ins. H. Poincaré Anal. Non Linéaire, 27 (2010), 1471–1487. https://doi.org/10.1016/j.anihpc.2010.09.004 doi: 10.1016/j.anihpc.2010.09.004
[18]	J. Korvenpää, T. Kuusi, E. Lindgren, Equivalence of solutions to fractional $p$-Laplace type equations, J. Math. Pure. Appl., 132 (2019), 1–26. https://doi.org/10.1016/j.matpur.2017.10.004 doi: 10.1016/j.matpur.2017.10.004
[19]	M. Medina, P. Ochoa, On viscosity and weak solutions for non-homogeneous p-Laplace equations, Adv. Nonlinear Anal., 8 (2019), 468–481. https://doi.org/10.1515/anona-2017-0005 doi: 10.1515/anona-2017-0005
[20]	J. Siltakoski, Equivalence of viscosity solutions and weak solutions for the normalized $p(x)$-Laplacian, Calc. Var., 57 (2018), 95. https://doi.org/10.1007/s00526-018-1375-1 doi: 10.1007/s00526-018-1375-1
[21]	P. Takac, J. Giacomoni, A $p(x)$-Laplacian extension of the Díaz-Saa inequality and some applications, P. Roy. Soc. Edinb. A, 150 (2020), 205–232. https://doi.org/10.1017/prm.2018.91 doi: 10.1017/prm.2018.91

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