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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 \(\partial M\).

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
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