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Представления второго рода для решений соболевских классов общей линейной эллиптической системы первого порядка в односвязной плоской области

Authors:
S. B. Klimentov
Sibirskii Matematicheskii Zhurnal, 2021, 62:3, 538–554
References
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Full-text available
The aim of this paper is to prove the following version of well-known Kellogg’s theorem. Let , , and is a conformal (one-to-one, onto) mapping. Then, extends to the homeomorphism from to ; moreover, , and the inverse mapping .
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We consider a second kind representation for solutions to a first order general uniformly elliptic linear system in a simply connected plane domain G with the Wpk1p W^{k-\frac{1}{p}}_{p} -boundary. We prove that the operator of the system is an isomorphism of Sobolev’s space Wpk(G) W^{k}_{p}(\overline{G}) , k1 k\geq 1 , p>2 p>2 , under appropriate assumptions about coefficients and the boundary. These results are new even for solutions to the canonical first order elliptic system (generalized analytic functions in the sense of Vekua).
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There has been a great deal of work done in recent years on weighted Bergman spaces A 1, while others require substantial additional effort. Some of our results about integral representations, complex in- terpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.
Obobschennye resheniya sistemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa s razryvnymi koeffitsientami
  • B V Boyarskii
Boyarskii B. V., "Obobschennye resheniya sistemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa s razryvnymi koeffitsientami", Mat. sb., 43:4 (1957), 451-503 Math-Net. Ru
Ob izomorfnosti nekotorykh funktsionalnykh prostranstv pri deistvii integrodifferentsialnykh operatorov
  • S B Klimentov
Klimentov S. B., "Ob izomorfnosti nekotorykh funktsionalnykh prostranstv pri deistvii integrodifferentsialnykh operatorov", Ufim. mat. zhurn., 11:1 (2019), 39-60 Math-Net. Ru Zentralblatt MATH
Otsenki reshenii ellipticheskikh uravnenii vblizi granitsy, Izd-vo inostr
  • S Agmon
  • A Duglis
  • L Nirenberg
Agmon S., Duglis A., Nirenberg L., Otsenki reshenii ellipticheskikh uravnenii vblizi granitsy, Izd-vo inostr. lit., M., 1962
Kraevye zadachi so svobodnymi granitsami dlya ellipticheskikh sistem uravnenii
  • V N Monakhov
Monakhov V. N., Kraevye zadachi so svobodnymi granitsami dlya ellipticheskikh sistem uravnenii, Nauka, Novosibirsk, 1977