Abstract
We prove the higher integrability of the gradient of solutions of the Zaremba problem in a bounded strongly Lipschitz domain for an inhomogeneous \(p(\,\cdot\,)\)-Laplace equation with a variable exponent \(p\) having a logarithmic continuity modulus.
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References
V. V. Zhikov, On Variational Problems and Nonlinear Elliptic Equations with Nonstandard Growth Conditions [in Russian], Tamara Rozhkovskaya, Novosibirsk (2017).
V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Math. USSR-Izv., 29, 33–66 (1987).
V. V. Zhikov, “On Lavrentiev’s phenomenon,” Russ. J. Math. Phys., 3, 249–269 (1995).
V. V. Zhikov and S. E. Pastukhova, “Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent,” Sb. Math., 199, 1751–1782 (2008).
L. Diening, P. Harjulehto, P. Hästö, and M. R\(\mathring{u}\)žička, Lebesgue and Sobolev Spaces with Variable Exponents (Lecture Notes in Mathematics, Vol. 2017), Springer, Heidelberg (2011).
J.-P. Lions, Some Methods for Solving Nonlinear Boundary-Value Problems, Dunod-Gauthier-Villars, Paris (1969).
D. Kinderlerer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980).
B. V. Boyarskii, “Generalized soluions of a system of differential equations of the first order of elliptic type with discontinuous coefficients [in Russian],” Mat. Sb. (N.S.), 43(85), 451–503 (1957).
N. G. Meyers, “An \(L^p\)-estimate for the gradient of solutions of second order elliptic deivergence equations,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17, 189–206 (1963).
Yu. A. Alkhutov, G. A. Chechkin, and V. G. Maz’ya, “On the Boyarsky–Meyers estimate of a solution to the Zaremba Problem,” Arch. Ration. Mech. Anal., 245, 1197–1211 (2022).
Yu. A. Alkhutov and A. G. Chechkina, “Many-dimensional Zaremba problem for an inhomogeneous \(p\)-Laplace equation,” Dokl. Math., 106, 243–246 (2022).
E. Acerbi and G. Mingione, “Gradient estimates for the \(p(x)\)-Laplacian system,” J. Reine Angew. Math., 584, 117–148 (2005).
L. Diening and S. Schwarzsacher, “Global gradient estimates for the \(p(\cdot)\)-Laplacian,” Nonlinear Anal., 106, 70–85 (2014).
V. V. Zhikov, “On some variational problems,” Russ. J. Math. Phys., 5, 105–116 (1997).
G. Cimatti and G. Prodi, “Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor,” Ann. Matem. Pura Appl., 152, 227–236 (1988).
S. D. Howison, J. F. Rodriges, and M. Shillor, “Stationary solutions to the thermistor problem,” J. Math. Anal. Appl., 174, 573–588 (1993).
M. R\(\mathring{u}\)žička, Electrorheological Fluids: Modeling and Mathematical Theory (Lecture Notes in Mathematics, Vol. 1748), Springer, Berlin–Heidelberg (2000).
V. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations (Grundlehren der mathematischen Wissenschaften, Vol. 342), Springer, Berlin–Heidelberg (2011).
F. W. Gehring, “The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping,” Acta Math., 130, 265–277 (1973).
M. Giaquinta and G. Modica, “Regularity results for some classes of higher order nonlinear elliptic systems,” J. für die reine und angewandte Math., 1979, 145–169 (1979).
I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems (Translations of Mathematical Monographs, Vol. 139), AMS, Providence, RI (1994).
Funding
The results in Sec. 3 belong to the first author. The proof of Lemma 1 and Theorem 1 of that section was supported by a grant from the Russian Science Foundation (project No. 22-21-00292), and the proof of Theorem 2 was supported by the State Assignment of the Vladimir State University (FZUN-2023-0004). The results of the second author in Sec. 2 were supported by a grant of the Russian Science Foundation (project No. 20-11-20272); and those in Sec. 1, by the grant from the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (project AP14869553).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 218, pp. 3–22 https://doi.org/10.4213/tmf10522.
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Alkhutov, Y.A., Chechkin, G.A. Multidimensional Zaremba problem for the \(p(\,\cdot\,)\)-Laplace equation. A Boyarsky–Meyers estimate. Theor Math Phys 218, 1–18 (2024). https://doi.org/10.1134/S004057792401001X
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DOI: https://doi.org/10.1134/S004057792401001X