@article{SM_2023_214_9_a3,
author = {A. G. Chechkina},
title = {On the {Zaremba} problem for the $p$-elliptic equation},
journal = {Sbornik. Mathematics},
pages = {1321--1336},
year = {2023},
volume = {214},
number = {9},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2023_214_9_a3/}
}
A. G. Chechkina. On the Zaremba problem for the $p$-elliptic equation. Sbornik. Mathematics, Tome 214 (2023) no. 9, pp. 1321-1336. http://geodesic.mathdoc.fr/item/SM_2023_214_9_a3/
[1] B. V. Bojarski, “Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients”, Mat. Sb., 43(85):4 (1957), 451–503 (Russian) | MR | Zbl
[2] N. G. Meyers, “An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17:3 (1963), 189–206 | MR | Zbl
[3] V. V. Zhikov, “On some variational problems”, Russian J. Math. Phys., 5:1 (1997), 105–116 | MR | Zbl
[4] E. Acerbi and G. Mingione, “Gradient estimates for the $p(x)$-Laplacian system”, J. Reine Angew. Math., 2005:584 (2005), 117–148 | DOI | MR | Zbl
[5] L. Diening and S. Schwarzsacher, “Global gradient estimates for the $p(\cdot)$-Laplacian”, Nonlinear Anal., 106 (2014), 70–85 | DOI | MR | Zbl
[6] G. Cimatti and G. Prodi, “Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor”, Ann. Mat. Pura Appl. (4), 152 (1988), 227–236 | DOI | MR | Zbl
[7] S. D. Howison, J. F. Rodrigues and M. Shillor, “Stationary solutions to the thermistor problem”, J. Math. Anal. Appl., 174:2 (1993), 573–588 | DOI | MR | Zbl
[8] S. Zaremba, “Sur un problème mixte relatif à l'équation de Laplace”, Bull. Acad. Sci. Cracovie. Cl. Sci. Math. Nat. Ser. A, 1910, 313–344 | Zbl
[9] G. Fichera, “Sul problema misto per le equazioni lineari alle derivate parziali del secondo ordine di tipo ellittico”, Rev. Roumaine Math. Pures Appl., 9 (1964), 3–9 | MR | Zbl
[10] V. G. Maz'ya, “Some estimates for solutions of elliptic second-order equations”, Soviet Math. Dokl., 2 (1961), 413–415 | MR | Zbl
[11] V. V. Zhikov and S. E. Pastukhova, “Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent”, Sb. Math., 199:12 (2008), 1751–1782 | DOI | MR | Zbl
[12] M. Giaquinta and G. Modica, “Regularity results for some classes of higher order non linear elliptic systems”, J. Reine Angew. Math., 311/312 (1979), 145–169 | MR | Zbl
[13] Yu. A. Alkhutov and G. A. Chechkin, “Increased integrability of the gradient of the solution to the Zaremba problem for the Poisson equation”, Dokl. Math., 103:2 (2021), 69–71 | DOI | MR | Zbl
[14] Yu. A. Alkhutov and G. A. Chechkin, “The Meyer's estimate of solutions to Zaremba problem for second-order elliptic equations in divergent form”, C. R. Mécanique, 349:2 (2021), 299–304 | DOI
[15] Yu. A. Alkhutov, G. A. Chechkin and V. G. Maz'ya, “Boyarsky-Meyers estimate for solutions to Zaremba problem”, Arch. Ration. Mech. Anal., 245:2 (2022), 1197–1211 | DOI | MR | Zbl
[16] Yu. A. Alkhutov and A. G. Chechkina, “Many-dimensional Zaremba problem for an inhomogeneous $p$-Laplace equation”, Dokl. Math., 106:1 (2022), 243–246 | DOI | MR | Zbl
[17] G. A. Chechkin, “The Meyers estimates for domains perforated along the boundary”, Mathematics, 9:23 (2021), 3015, 11 pp. | DOI
[18] V. G. Maz'ja, Sobolev spaces, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1985, xix+486 pp. | DOI | MR | Zbl
[19] G. A. Chechkin, Yu. O. Koroleva and L.-E. Persson, “On the precise asymptotics of the constant in Friedrich's inequality for functions vanishing on the part of the boundary with microinhomogeneous structure”, J. Inequal. Appl., 2007 (2007), 34138, 13 pp. | DOI | MR | Zbl
[20] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969, xx+554 pp. | MR | Zbl
[21] G. I. Laptev, “Monotonicity conditions for a class of quasilinear differential operators depending on parameters”, Math. Notes, 96:3 (2014), 379–390 | DOI | MR | Zbl
[22] M. D. Surnachev and V. V. Zhikov, “On existence and uniqueness classes for the Cauchy problem for parabolic equations of the $p$-Laplace type”, Commun. Pure Appl. Anal., 12:4 (2013), 1783–1812 | DOI | MR | Zbl
[23] V. G. Maz'ya, “The continuity at a boundary point of the solutions of quasi-linear elliptic equations”, Vestn. Leningrad. Univ., 1970, no. 13, 42–55 (Russian) | MR | Zbl
[24] F. W. Gehring, “The $L^p$-integrability of the partial derivatives of a quasiconformal mapping”, Acta Math., 130 (1973), 265–277 | DOI | MR | Zbl
[25] I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Transl. Math. Monogr., 139, Amer. Math. Soc., Providence, RI, 1994, xii+348 pp. | MR | MR | Zbl | Zbl
[26] D. R. Adams and N. G. Meyers, “Thinness and Wiener criteria for non-linear potentials”, Indiana Univ. Math. J., 22:2 (1972), 169–197 | DOI | MR | Zbl
[27] D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Grundlehren Math. Wiss., 314, Springer-Verlag, Berlin, 1996, xii+366 pp. | DOI | MR | Zbl
[28] T. Sjödin, “Capacities of compact sets in linear subspaces of $R^n$”, Pacific J. Math., 78:1 (1978), 261–266 | DOI | MR | Zbl
[29] V. G. Maz'ya and V. P. Khavin, “A nonlinear analogue of the Newtonian potential and metric properties of the $(p,\ell)$-capacity”, Soviet Math. Dokl., 11 (1970), 1294–1298 | MR | Zbl
[30] V. G. Maz'ya and V. P. Khavin, “Non-linear potential theory”, Russian Math. Surveys, 27:6 (1972), 71–148 | DOI | MR | Zbl
[31] G. A. Chechkin, “Averaging of boundary value problems with a singular perturbation of the boundary conditions”, Russian Acad. Sci. Sb. Math., 79:1 (1994), 191–222 | DOI | MR | Zbl
[32] G. A. Chechkin and R. R. Gadyl'shin, “On boundary-value problems for the Laplacian in bounded domains with micro inhomogeneous structure of the boundaries”, Acta Math. Sin. (Engl. Ser.), 23:2 (2007), 237–248 | DOI | MR | Zbl
[33] G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and applications, Transl. Math. Monogr., 234, Amer. Math. Soc., Providence, RI, 2007, x+234 pp. | DOI | MR | Zbl