Abstract
The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in \(L_{2+\delta }\), \(\delta >0\), for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero \((n-1)\)-dimensional measure and non-zero p-capacity, \(p>1\) is constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
Code availability
Not applicable.
References
Chechkin, G.A.: On boundary value problems for a second-order elliptic equation with oscillating boundary conditions (Russian). In: Vragov, V.N. (ed.) Nonclassical Partial Differential Equations, pp. 95–104. IM SOAN SSSR Press, Novosibirsk, 1988
Chechkin, G.A., Piatnitski, A.L., Shamaev, A.S.: Homogenization. Methods and Applications. Translations of Mathematical Monographs, vol. 234. AMS, Providence, 2007
Zaremba, S.: Sur un problème mixte relatif à l’équation de Laplace (French). Bulletin de l’Académie des sciences de Cracovie, Classe des sciences mathématiques et naturelles, serie A, 313–344, 1910
Fichera, G.: Sul problema misto per le equazioni lineari alle derivate parziali del secondo ordine di tipo ellittico (Italian). Rev. Roumaine Math. Pures Appl. 9, 3–9, 1964
Maz’ya, V.G.: Some estimates of solutions of second-order elliptic equations (Russian). Dokl. Akad. Nauk SSSR 137(5), 1057–1059, 1961
Kerimov, T.M., Maz’ya, V.G., Novruzov, A.A.: A criterion for the regularity of the infinitely distant point for the Zaremba problem in a half-cylinder (Russian). Z. Anal. Anwendungen 7(2), 113–125, 1988
Maz’ya, V.G.: Boundary Behavior of Solutions to Elliptic Equations in General Domains. EMS Tracts in Mathematics, vol. 30. EMS Publishing House, Zürich, 2018
Bojarski, B.V.: Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients (Russian). Mat. Sb. N.S. 43(85), 451–503, 1957
Meyers, N.G.: An \(L^p\)-estimate for the gradient of solutions of second order elliptic deivergence equations. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3-e série 17(3), 189–206, 1963
Zhikov, V.V.: On some variational problems. Russian J. Math. Phys. 5(1), 105–116, 1997
Breit, D., Cianchi, A., Diening, L., Kuusi, T., Schwarzacher, S.: Pointwise Calderón-Zygmund gradient estimates for the \(p\)-Laplace system. J. Math. Pures Appl. 114, 146–190, 2018
Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacian system. J. Reine Angew. Math. 584, 117–148, 2005
Diening, L., Schwarzsacher, S.: Global gradient estimates for the \(p(\cdot )\)-Laplacian. Nonlinear Anal. 106, 70–85, 2014
Alkhutov, Yu.A., Chechkin, G.A.: Increased integrability of the gradient of the solution to the Zaremba problem for the Poisson equation. Russian Academy of Sciencies. Doklady Mathematics 103(2), 69–71, 2021 (Translated from Doklady Akademii Nauk 497(2), 3–6, 2021)
Chechkin, G.A.: The Meyers estimates for domains perforated along the boundary. Mathematics 9(23), Art number 3015, 2021
Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer-Verlag, Berlin, 2011
Caccioppoli, R.: Limitazioni integrali per le soluzioni di un’equazione lineare ellitica a derivate parziali (Italian). Giorn. Mat. Battaglini 80, 186–212, 1950–1951
Gehring, F.W.: The \(L_p\)-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277, 1973
Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. Journ. für die reine und angewandte Math. 311/312, 145–169, 1979
Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Translations of Math. Monographs, vol. 139. AMS, Providence, 1994
Il’in, V.P.: On a imbedding theorem for limiting exponent (Russian). Dokl. Akad. Nauk SSSR 96, 905–908, 1954
Adams, D.R., Meyers, N.G.: Thinness and Wiener criteria for nonlinear potentials. Indiana Univ. Math. J. 22, 139–158, 1972
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin, 1996
Sjödin, T.: Capacities of compact sets in linear subspaces of \(\mathbb{R}^n\). Pac. J. Math. 78, 261–266, 1978
Maz’ya, V.G., Havin, V.P.: A nonlinear analogue of the Newtonian potential, and metric properties of \((p,\, l)\)-capacity (Russian). Dokl. Akad. Nauk SSSR 194, 770–773, 1970
Maz’ya, V.G., Havin, V.P.: Nonlinear potential theory (Russian). Russ. Math. Surv. 27, 71–148, 1972 (Translated from Usp. Mat. Nauk 27, 67–138, 1972)
Acknowledgements
The work of Yurij A. Alkhutov in Section 3 was supported in part by RSF (Project 22-21-00292). The work of Gregory A. CHECHKIN in Section 2 was supported in part by RSF (Project 20-11-20272). The Vladimir G. Maz’ya was supported by the RUDN University Strategic Academic Leadership Program.
Funding
RSF (project 22-21-00292), RSF (project 20-11-20272), RUDN University Strategic Academic Leadership Program
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
Not applicable.
Ethics approval
Not applicable.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
Additional information
Communicated by D. Kinderlehrer.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Alkhutov, Y.A., Chechkin, G.A. & Maz’ya, V.G. Boyarsky–Meyers Estimate for Solutions to Zaremba Problem. Arch Rational Mech Anal 245, 1197–1211 (2022). https://doi.org/10.1007/s00205-022-01805-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-022-01805-0