Anisotropic problem with non-local boundary conditions and measure data

A. Kaboré; S. Ouaro

doi:10.4067/S0719-06462021000100021

DOI:

https://doi.org/10.4067/S0719-06462021000100021

Abstract

We study a nonlinear anisotropic elliptic problem with non-local boundary conditions and measure data. We prove an existence and uniqueness result of entropy solution.

Keywords

Entropy solution , non-local boundary conditions , Leray-Lions operator , bounded Radon diffuse measure , Marcinkiewicz spaces

A. Kaboré Laboratoire de Mathematiques et Informatiques (LAMI), UFR. Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso.
S. Ouaro Laboratoire de MathemÃ tiques et Informatiques (LAMI), UFR. Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso.

Pages: 21–62
Date Published: 2021-04-13
Vol. 23 No. 1 (2021)

A. Baalal, and M. Berghout, “The Dirichlet problems for nonlinear elliptic equations with variable exponent”, Journal of Applied Analysis and Computation, vol. 9, no. 1, pp. 295–313, 2019.

M. B. Benboubker, H. Hjiaj, and S. Ouaro, “Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent”, Journal of Applied Analysis and Computation, vol. 4, no. 3, pp. 245–270, 2014.

M. Bendahmane, and K. H. Karlsen, “Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps into spheres”, Electron. J. Differential Equations, no. 46, 30 pp, 2006.

Ph. Bénilan, H. Brézis, and M. G. Crandall, “A semilinear equation in L1(RN )”. Ann. Scuola. Norm. Sup. Pisa, vol. 2, pp. 523–555, 1975.

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vazquez, “An L1 theory of existence and uniqueness of nonlinear elliptic equations”, Ann Sc. Norm. Super. Pisa, vol. 22, no. 2, pp. 240–273, 1995.

B. K. Bonzi, S. Ouaro, and F. D. Y. Zongo, “Entropy solution for nonlinear elliptic anisotropic homogeneous Neumann Problem”, Int. J. Differ. Equ, Article ID 476781, 2013.

M. M. Boureanu, and V. D. Radulescu, “Anisotropic Neumann problems in Sobolev spaces with variable exponent”. Nonlinear Anal. TMA, vol. 75, no. 12, pp. 4471–4482, 2012.

B. Koné, S. Ouaro and F. D. Y. Zongo, “Nonlinear elliptic anisotropic problem with Fourier boundary condition”, Int. J. Evol. Equ, vol. 8, no 4, pp. 305–328, 2013.

Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM J. Appl. Math, vol. 66, pp. 1383–1406, 2006.

L. Diening, “Theoretical and Numerical Results for Electrorheological Fluids”, PhD. thesis, University of Frieburg, Germany, 2002.

L. Diening, “Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(.) and W1,p(.)”, Math. Nachr, vol. 268, pp. 31–43, 2004.

Y. Ding, T. Ha-Duong, J. Giroire, and V. Mouma, “Modeling of single-phase flow for horizontal wells in a stratified medium”, Computers and Fluids, vol. 33, pp. 715–727, 2004.

X. Fan, and D. Zhao, “On the spaces Lp(.)(Î©) and Wm,p(.)(Î©)”, J. Math. Anal. Appl., vol. 263, pp. 424–446, 2001.

X. Fan, “Anisotropic variable exponent Sobolev spaces and p(.) -Laplacian equations”, Complex variables and Elliptic Equations. vol. 55, pp. 1–20, 2010.

J. Giroire, T. Ha-Duong, and V. Moumas, “A non-linear and non-local boundary condition for a diffusion equation in petroleum engineering”, Mathematical Methods in the Applied Sciences, vol. 28, no. 13, pp. 1527–1552, 2005.

P. Halmos, Measure Theory, D. Van Nostrand Company, New York, 1950.

T.C. Halsey, “Electrorheological fluids”, Science, vol. 258, ed. 5083, pp. 761–766, 1992.

H. Hudzik, “On generalized Orlicz-Sobolev space”, Funct. Approximatio Comment. Math., vol. 4, pp. 37–51, 1976.

I. Ibrango, and S. Ouaro, “Entropy solutions for nonlinear Dirichlet problems”, Annals of the university of craiova, Mathematics and Computer Science Series, vol. 42, no. 2, pp. 347–364, 2015.

I. Ibrango, and S. Ouaro, “Entropy solutions for nonlinear elliptic anisotropic problems with homogeneous Neumann boundary condition”, Journal of Applied Analysis and Computation, vol. 6, no. 2, pp. 271–292, 2016.

A. Kaboré, and S. Ouaro, “Nonlinear Elliptic anisotropic problem involving non local boundary conditions with variable exponent and graph data”, Creative Mathematics, vol. 29, no. 2, pp. 145–152, 2020.

I. Konaté, and S. Ouaro, “Good Radon measure for anisotropic problems with variable exponent” Electron. J. Diff Equ., vol. 2016, no. 221, pp. 1–19, 2016.

O. Kovacik, and J. Rakosnik, “On spaces Lp(x) and W1,p(x)”, Czech. Math. J., vol. 41, pp. 592–618, 1991.

L. M. Kozhevnikova, “On solutions of elliptic equations with variable exponents and measure data in Rn”, 2019. arXiv : 1912.12432.

L. M. Kozhevnikova, “On solutions of anisotropic elliptic equations with variable exponent and measure data”, Complex Variables and Elliptic Equations, vol. 65, no. 3, pp. 333–367, 2020.

M. Mihailescu, and V. Radulescu, “A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., vol. 462, pp. 2625–2641, 2006.

M. Mihailescu, and V. Radulescu, “On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent”, Proc. Amer. Math. Soc., vol. 135, pp. 2929–2937, 2007.

M. Mihailescu, P. Pucci, and V. Radulescu, “Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent”, J. Math. Anal. Appl., vol. 340, no. 1, pp. 687–698, 2008.

J. Musielak, Orlicz Spaces, and modular spaces, Lecture Notes in Mathematics, Springer, Berlin, 1983.

H. Nakano, “Modulared semi-ordered linear spaces”, Tokyo: Maruzen Co. Ltd, 1950.

I. Nyanquini, S. Ouaro, and S. Safimba, “Entropy solution to nonlinear multivalued elliptic problem with variable exponents and measure data”. Ann. Univ. Craiova ser. Mat. Inform., vol. 40, no.2, pp. 174–198, 2013.

W. Orlicz, “Ìber konjugierte Exponentenfolgen”, Studia Math., vol. 3, pp. 200–212, 1931.

C. Pfeiffer, C. Mavroidis, Y. Bar-Cohen, and B. Dolgin, “Electrorheological fluid based force feedback device”, in Proc. 1999 SPIE Telemanipulator and Telepresence Technologies VI Conf. (Boston, MA), vol. 3840, pp. 88–99, 1999.

V. Radulescu, “Nonlinear elliptic equations with variable exponent: old and new”, Nonlinear Anal., vol. 121, pp. 336–369, 2015.

K.R. Rajagopal, and M. Ruzicka, “Mathematical modelling of electrorheological fluids”, Continuum Mech. Thermodyn., vol. 13, pp. 59–78, 2001.

M. Ruzicka, Electrorheological fluids: modelling and mathematical theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000.

M. Sanchon, and J. M. Urbano, “Entropy solutions for the p(x)-Laplace Equation”, Trans. Amer. Math. Soc., vol. 361, no. 12, pp. 6387–6405, 2009.

U. Sert, and K. Soltanov, “On the solvability of a class of nonlinear elliptic type equation with variable exponent”, Journal of Applied Analysis and Computation, vol. 7, no. 3, pp. 1139–1160, 2019.

M. Troisi, “Teoremi di inclusione per spazi di Sobolev non isotropi”. Ric. Mat., vol. 18, pp. 3–24, 1969.

I. Sharapudinov, “On the topology of the space Lp(t)([0,1])”, Math. Zametki, vol. 26, pp. 613–632, 1978.

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, vol. 49, 1997.

I.V. Tsenov, “Generalization of the problem of best approximation of a function in the space Ls”, Uch. Zap. Dagestan Gos. Univ., vol. 7, pp. 25–37, 1961.

W. M. Winslow, “Induced Fibration of Suspensions”, J. Applied Physics, vol. 20, pp. 1137–1140, 1949.