Behavior of solutions of the Dirichlet Problem for the 𝑝(𝑥)-Laplacian at a boundary point

Alkhutov, Yu.; Surnachev, M.

doi:10.1090/spmj/1595

Behavior of solutions of the Dirichlet Problem for the $p(x)$-Laplacian at a boundary point
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by Yu. A. Alkhutov and M. D. Surnachev
Translated by: E. Peller

St. Petersburg Math. J. 31 (2020), 251-271

DOI: https://doi.org/10.1090/spmj/1595

Published electronically: February 4, 2020

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Abstract:

The Dirichlet problem for the $p(x)$-Laplacian with a continuous boundary function is treated. A sufficient condition is indicated for the regularity of a boundary point, and the modulus of continuity of solutions at this point is estimated.

References

V. V. Zhikov, Questions of convergence, duality and averaging for functionals of the calculus of variations, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 5, 961–998 (Russian). MR 718413
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877 (Russian). MR 864171
V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system, Differ. Uravn. 33 (1997), no. 1, 107–114, 143 (Russian, with Russian summary); English transl., Differential Equations 33 (1997), no. 1, 108–115. MR 1607245
Michael Růžička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. MR 1810360, DOI 10.1007/BFb0104029
E. Acerbi, G. Mingione, and G. A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 1, 25–60 (English, with English and French summaries). MR 2037246, DOI 10.1016/S0294-1449(03)00031-3
V. V. Zhikov, Solvability of the three-dimensional thermistor problem, Tr. Mat. Inst. Steklova 261 (2008), no. Differ. Uravn. i Din. Sist., 101–114 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 261 (2008), no. 1, 98–111. MR 2489700, DOI 10.1134/S0081543808020090
Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR 2790542, DOI 10.1007/978-3-642-18363-8
Vasiliĭ V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3 (1995), no. 2, 249–269. MR 1350506
V. V. Zhikov, On setting boundary value problems for integrants type $|\xi |^{\alpha (x)}$, In Moscow Mathematical Society, Uspelki Mat. Nauk 41 (1986), no. 4, 187–188. (Russian)
Vasiliĭ V. Zhikov, On some variational problems, Russian J. Math. Phys. 5 (1997), no. 1, 105–116 (1998). MR 1486765
Yu. A. Alkhutov and O. V. Krasheninnikova, Continuity at boundary points of solutions of quasilinear elliptic equations with a nonstandard growth condition, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 6, 3–60 (Russian, with Russian summary); English transl., Izv. Math. 68 (2004), no. 6, 1063–1117. MR 2108520, DOI 10.1070/IM2004v068n06ABEH000509
Yu. A. Alkhutov and M. D. Surnachev, Regularity of a boundary point for the $p(x)$-Laplacian, J. Math. Sci. (N.Y.) 232 (2018), no. 3, Problems in mathematical analysis. No. 92 (Russian), 206–231. MR 3813478, DOI 10.1007/s10958-018-3870-5
Yu. A. Alkhutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition, Differ. Uravn. 33 (1997), no. 12, 1651–1660, 1726 (Russian, with Russian summary); English transl., Differential Equations 33 (1997), no. 12, 1653–1663 (1998). MR 1669915
Oskar Perron, Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u=0$, Math. Z. 18 (1923), no. 1, 42–54 (German). MR 1544619, DOI 10.1007/BF01192395
N. Wiener, Certain notions in potential theory, J. Math. Phys. 3 (1924), no. 1, 24–51.
—, The Dirichlet problem, J. Math. Phys. 3 (1924), no. 3, 127–146.
—, Note on a paper of O. Perron, J. Math. Phys. 4 (1925), no. 1–4, 21–32.
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
V. A. Kondrat′ev and E. M. Landis, Qualitative theory of second-order linear partial differential equations, Partial differential equations, 3 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 99–215, 220 (Russian). MR 1133457, DOI 10.1134/S0081543814050101
H. L. Lebesgue, Sur des cas d’impossibilité du problème de Dirichlet, C. R. Soc. Math. France 41 (1913), 17.
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
V. G. Maz′ja, On the modulus of continuity of a solution of the Dirichlet problem near an irregular boundary, Problems Math. Anal. Boundary Value Problems Integr. Equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1966, pp. 45–58 (Russian). MR 0211047
V. G. Maz′ja, The behavior near the boundary of the solution of the Dirichlet problem for an elliptic equation of the second order in divergence form, Mat. Zametki 2 (1967), 209–220 (Russian). MR 219873
V. G. Maz′ja, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ. 25 (1970), no. 13, 42–55 (Russian, with English summary). MR 0274948
I. N. Krol′ and V. G. Maz′ja, The absence of the continuity and Hölder continuity of the solutions of quasilinear elliptic equations near a nonregular boundary, Trudy Moskov. Mat. Obšč. 26 (1972), 75–94 (Russian). MR 0377265
Ronald Gariepy and William P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 67 (1977), no. 1, 25–39. MR 492836, DOI 10.1007/BF00280825
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), no. 3-4, 153–171. MR 806413, DOI 10.1007/BF02392541
Tero Kilpeläinen and Jan Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137–161. MR 1264000, DOI 10.1007/BF02392793
I. I. Sharapudinov, Some questions of approximation theory in Lebesgue spaces with a variable exponent, UMI VNC RAN i RSO-A, Vladikavkaz, 2012. (Russian)
Neil S. Trudinger, On the regularity of generalized solutions of linear, non-uniformly elliptic equations, Arch. Rational Mech. Anal. 42 (1971), 50–62. MR 344656, DOI 10.1007/BF00282317