Geodesic
On the Boyarsky--Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift
Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 1-14.

Voir la notice de l'article provenant de la source Math-Net.Ru

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.
Keywords: Zaremba problem, Meyers estimates, embedding theorems, increased integrability. 
@article{CMFD_2024_70_1_a0,
     author = {Yu. A. Alkhutov and G. A. Chechkin},
     title = {On the {Boyarsky--Meyers} estimate for the solution of the {Dirichlet} problem for a second-order linear elliptic equation with drift},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {1--14},
     publisher = {mathdoc},
     volume = {70},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a0/}
}
TY  - JOUR
AU  - Yu. A. Alkhutov
AU  - G. A. Chechkin
TI  - On the Boyarsky--Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2024
SP  - 1
EP  - 14
VL  - 70
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a0/
LA  - ru
ID  - CMFD_2024_70_1_a0
ER  - 
%0 Journal Article
%A Yu. A. Alkhutov
%A G. A. Chechkin
%T On the Boyarsky--Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift
%J Contemporary Mathematics. Fundamental Directions
%D 2024
%P 1-14
%V 70
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a0/
%G ru
%F CMFD_2024_70_1_a0
Yu. A. Alkhutov; G. A. Chechkin. On the Boyarsky--Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift. Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 1-14. http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a0/

[1] Boyarskii B. V., “Obobschennye resheniya sistemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa s razryvnymi koeffitsientami”, Mat. sb., 43:4 (1957), 451–503

[2] Gilbarg D., Trudinger N. S., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR

[3] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[4] Chechkin G. A., Chechkina T. P., “Otsenka Boyarskogo—Meiersa dlya divergentnykh ellipticheskikh uravnenii vtorogo poryadka. Dva prostranstvennykh primera”, Probl. mat. analiza, 119 (2022), 107–116 | Zbl

[5] Chechkina A. G., “O zadache Zaremby dlya $p$-ellipticheskogo uravneniya”, Mat. sb., 214:9 (2023), 144–160 | DOI | MR

[6] Acerbi E., Mingione G., “Gradient estimates for the $p(x)$-Laplacian system”, J. Reine Angew. Math., 584 (2005), 117–148 | DOI | MR | Zbl

[7] Alkhutov Yu. A., Chechkin G. A., “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

[8] Alkhutov Yu. A., Chechkin G. A., “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

[9] Alkhutov Yu. A., Chechkin G. A., Maz'ya V. G., “On the Bojarski–Meyers estimate of a solution to the Zaremba problem”, Arch. Ration. Mech. Anal., 245:2 (2022), 1197–1211 | DOI | MR | Zbl

[10] Chechkin G. A., “The Meyers estimates for domains perforated along the boundary”, Mathematics, 9:23 (2021) | DOI | Zbl

[11] Cimatti G., Prodi G., “Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor”, Ann. Mat. Pura Appl., 63 (1988), 227–236 | DOI | MR

[12] Diening L., Schwarzsacher S., “Global gradient estimates for the $p(\cdot)$-Laplacian”, Nonlinear Anal., 106 (2014), 70–85 | DOI | MR | Zbl

[13] Gehring F. W., “The $L_p$-integrability of the partial derivatives of a quasiconformal mapping”, Acta Math., 130 (1973), 265–277 | DOI | MR | Zbl

[14] Giaquinta M., Modica G., “Regularity results for some classes of higher order nonlinear elliptic systems”, J. Reine Angew. Math., 311/312 (1979), 145–169 | MR | Zbl

[15] Howison S. D., Rodriges J. F., Shillor M., “Stationary solutions to the thermistor problem”, J. Math. Anal. Appl., 174 (1993), 573–588 | DOI | MR | Zbl

[16] Lax P. D., Milgram A., “Parabolic equations”, Contributions to the Theory of Partial Differential Equations, Princeton Univ. Press, Princeton, 1954, 167–190 | MR

[17] Meyers N. G., “An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations”, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17:3 (1963), 189–206 | MR | Zbl

[18] Skrypnik I. V., Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, AMS, Providence, 1994 | MR | Zbl

[19] Zhikov V. V., “On some variational problems”, Russ. J. Math. Phys., 5:1 (1997), 105–116 | MR | Zbl