Geodesic
Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 5-17 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Existence of a weak solution of the Dirichlet problem to nondiagonal elliptic systems with quadratic growth nonlinearities is proved in the two-dimensional case. It is established that the solution is smooth in the closure of a given domain with exception of at most finitely many points. The result is essentially based upon the theorem on “quasireverse” Hölder inequalities earlier proved by the author. 
@article{ZNSL_2003_295_a0,
     author = {A. A. Arkhipova},
     title = {Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--17},
     year = {2003},
     volume = {295},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a0/}
}
TY  - JOUR
AU  - A. A. Arkhipova
TI  - Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2003
SP  - 5
EP  - 17
VL  - 295
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a0/
LA  - ru
ID  - ZNSL_2003_295_a0
ER  - 
%0 Journal Article
%A A. A. Arkhipova
%T Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case)
%J Zapiski Nauchnykh Seminarov POMI
%D 2003
%P 5-17
%V 295
%U http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a0/
%G ru
%F ZNSL_2003_295_a0
A. A. Arkhipova. Solvability of nondiagonal elliptic systems with quadratic growth nonlinearities (two-dimensional case). Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 5-17. http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a0/

[1] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton, NJ, 1983 | MR

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

[3] S. Hildebrandt, H. Kaul, K.-O. Widman, “An existence theorem for harmonic mappings of Riemannian manifols”, Acta Math., 138 (1977), 1–16 | DOI | MR | Zbl

[4] J. Frehse, “On a class of nonlinear diagonal elliptic systems with critical growth and $\mathbb C^\alpha$-regularity”, Partial differential equations and the calculus of variations, II, Progr. Nonlinear Diff. Eq. Appl., 2, Birkhäuser, Boston, MA, 1989, 519–540 | MR

[5] R. Landes, “On the existence of weak solutions of perturbed systems with critical growth”, J. Reine Angew. Math., 393 (1989), 21–38 | DOI | MR | Zbl

[6] J. Frehse, “On two-dimensional quasi-linear elliptic systems”, Manuscripta Math., 28 (1979), 21–49 | DOI | MR | Zbl

[7] M. Struwe, Variational Methods, Springer-Verlag, 1990 | MR

[8] L. C. Evans, “Partial regularity for stationary harmonic maps into sphere”, Arch. Rat. Mech. Anal., 116 (1991), 101–113 | DOI | MR | Zbl

[9] T. Rivière, “Everywhere discontinuous harmonic maps into spheres”, Acta Math., 175 (1995), 197–226 | DOI | MR | Zbl

[10] A. A. Arkhipova, “O globalnoi razreshimosti zadachi Koshi–Dirikhle dlya nediagonalnykh parabolicheskikh sistem s variatsionnoi strukturoi pri dvukh prostranstvennykh peremennykh”, Sb. Problemy matem. analiza, vyp. 16, izd.S.-Peterb. un-ta, 1997, 3–40 | MR

[11] A. A. Arkhipova, “Lokalnaya i globalnaya po vremeni razreshimost zadachi Koshi–Dirikhle dlya klassa nelineinykh nediagonalnykh parabolicheskikh sistem”, Algebra i Analiz, 11:6 (1999), 69–102 | MR | Zbl

[12] A. Arkhipova, “Cauchy–Neumann problem for a class of nondiagonal parabolic systems withquadratic growth nonlinearities. I. On the continuability of smooth solutions”, Comment. Math. Univ. Carolonae, 41:4 (2000), 693–718 | MR | Zbl

[13] A. Arkhipova, “Cauchy–Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities. II. Local and global solvability results”, Comment. Math. Univ. Carolonae, 42:4 (2001), 53–76 | MR | Zbl

[14] A. Arkhipova, “Quasireverse Hölder inequalities and a priori estimates for strongly nonlinear systems”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 14:2 (2003), 91–108 | MR | Zbl