Geodesic
On the smoothness of solutions to elliptic equations in domains with~H\"older boundary
Eurasian mathematical journal, Tome 6 (2015) no. 3, pp. 76-92.

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

The dependence of the smoothness of variational solutions to the first boundary value problems for second order elliptic operators is studied. The results use Sobolev–Slobodetskii and Nikolskii–Besov spaces and their properties. Methods are based on the real interpolation technique and on generalization of the Savaré–Nirenberg difference quotient technique. 
@article{EMJ_2015_6_3_a5,
     author = {I. V. Tsylin},
     title = {On the smoothness of solutions to elliptic equations in domains {with~H\"older} boundary},
     journal = {Eurasian mathematical journal},
     pages = {76--92},
     publisher = {mathdoc},
     volume = {6},
     number = {3},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2015_6_3_a5/}
}
TY  - JOUR
AU  - I. V. Tsylin
TI  - On the smoothness of solutions to elliptic equations in domains with~H\"older boundary
JO  - Eurasian mathematical journal
PY  - 2015
SP  - 76
EP  - 92
VL  - 6
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2015_6_3_a5/
LA  - en
ID  - EMJ_2015_6_3_a5
ER  - 
%0 Journal Article
%A I. V. Tsylin
%T On the smoothness of solutions to elliptic equations in domains with~H\"older boundary
%J Eurasian mathematical journal
%D 2015
%P 76-92
%V 6
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2015_6_3_a5/
%G en
%F EMJ_2015_6_3_a5
I. V. Tsylin. On the smoothness of solutions to elliptic equations in domains with~H\"older boundary. Eurasian mathematical journal, Tome 6 (2015) no. 3, pp. 76-92. http://geodesic.mathdoc.fr/item/EMJ_2015_6_3_a5/

[1] O. V. Besov, V. P. Il'in, S. M. Nikolskii, Integral representations of functions and embedding theorems, Wiley, N.Y., 1978

[2] H. Brezis, Analyse fonctionnelle — Théorie et applications, Masson, Paris, 1983 | MR

[3] M. L. Gol'dman, “Imbedding theorems for anisotropic Nikol'skij Besov spaces with moduli of continuity of general form”, Proc. Steklov Inst. Math., 170, 1987, 95–116 | Zbl

[4] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, L., 1985 | MR | Zbl

[5] T. Kato, Perturbation theory for linear operators, Springer-Verlag, N.Y., 1966 | MR | Zbl

[6] D. Jerison, C. Kenig, “Boundary value problems on Lipschitz domains”, Studies in Partial Differential Equations, 23 (1982), 1–68 | MR | Zbl

[7] T. Muramatu, “On the dual of Besov spaces”, Publ. Res. Inst. Math. Kyoto Univ., 12, 1976, 123–140 | DOI | MR

[8] S. M. Nikolskii, Approximation of functions of several variables and embedding theorems, Army Foreign Science And Technology Center, Charlottesville, VA, 1971

[9] S. M. Nikolskii, “Properties of certain classes of functions of several variables on differentiable manifolds”, Math. sb., 33(75):2 (1953), 261–326 (in Russian) | MR

[10] V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains”, J. London Math. Soc., 60 (1999), 237–257 | DOI | MR | Zbl

[11] G. Savaré, “Regularity results for elliptic equations in Lipschitz domains”, J. Funct. Anal., 152 (1998), 176–201 | DOI | MR | Zbl

[12] G. Savaré, G. Schimperna, “Domain perturbations estimates for the solutions of second order elliptic equations”, J. Math. Pures Appl., 81:11 (2002), 1071–1112 | DOI | MR | Zbl

[13] A. M. Stepin, I. V. Tsylin, “On boundary value problems for elliptic operators in the case of domains on manifolds”, Doklady Mathematics, 92:1 (2015), 428–432 | DOI | Zbl

[14] H. Triebel, Theory of function spaces, Reprint of the 1983 Edition, Birkhauser, Basel, 2010 | MR