Storozhuk, K. V. On compact solvability of the exterior derivation for \(G\)-invariant boundary conditions. (English. Russian original) Zbl 0933.58001 Sib. Math. J. 40, No. 3, 585-589 (1999); translation from Sib. Mat. Zh. 40, No. 3, 683-688 (1999). Let \(M\) be a Riemannian manifold and let \(N\subset M\) be a compact submanifold which is a neighborhood about the boundary \(\partial M\) in \(M\). Denote by \(j\: L^k(N)\to L^k(M)\) the operator that extends by zero every form defined on \(N\) onto \(M\). Let \(\Gamma_M\) be a boundary value problem on \(M\). The corresponding boundary value problem on \(N\) is defined by putting \(\Gamma_N^k=\{\omega\in W_p^k(N)\mid j\omega\in\Gamma_M\}\). In the article under review, the author proves that, if the manifold \(M\) is compact, then the operator \[ d_{\Gamma_G}^k\: L_p^k(N)\to L_p^{k+1}(N) \] is compactly solvable for every \(p\in (1,\infty)\), where \(G\) is a group of isometries acting on \(\partial M\). Reviewer: V.Grebenev (Novosibirsk) Cited in 1 Document MSC: 58A10 Differential forms in global analysis 58A15 Exterior differential systems (Cartan theory) 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 58D10 Spaces of embeddings and immersions Keywords:exterior derivative; Riemannian manifold; compact solvability PDFBibTeX XMLCite \textit{K. V. Storozhuk}, Sib. Math. J. 40, No. 3, 683--688 (1999; Zbl 0933.58001); translation from Sib. Mat. Zh. 40, No. 3, 683--688 (1999) Full Text: DOI References: [1] V. M. Gol’dshteîn, V. I. Kuz’minov, and I. A. Shvedov, ”On normal and compact solvability of linear operators,” Sibirsk. Mat. Zh.,30, No. 5, 49–59 (1989). [2] V. M. Gol’dshteîn, V. I. Kuz’minov, and I. A. Shvedov, ”On normal and compact solvability of the operator of exterior derivation under homogeneous boundary conditions,” Sibirsk. Mat. Zh.,28, No. 4, 82–95 (1987). [3] V. I. Kuz’minov and I. A. Shvedov, ”On normal solvability of the operator of exterior derivation on a warped cylinder,” Sibirsk. Mat. Zh.,34, No. 6, 85–95 (1993). [4] V. I. Kuz’minov and I. A. Shvedov, ”On normal solvability of the operator of exterior derivation on warped products,” Sibirsk. Mat. Zh.,37, No. 2, 324–337 (1996). [5] V. I. Kuz’minov and I. A. Shvedov, ”On compact solvability of the operator of exterior derivation,” Sibirsk. Mat. Zh.,38, No. 3, 573–590 (1997). [6] Ya. A. Kopylov, ”On normal solvability of the operator of exterior derivation on a surface of revolution,” Sibirsk. Mat. Zh.,38, No. 6, 1300–1307 (1997). · Zbl 0920.58042 [7] N. Bourbaki, Integration: Vector Integration. Haar Measure. Convolution and Representations [Russian translation], Nauka, Moscow (1970). [8] N. Bourbaki, General Topology. Use of Real Numbers in General Topology. Function Spaces [Russian translation], Nauka, Moscow (1975). [9] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Publ., New York (1955). · Zbl 0068.01904 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.