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In this paper we consider the hypo-ellipticity of differential forms on a closed manifold.The main results show that there are some topological obstruct for the existence of the differential forms with hypoellipticity.
In order to characterise the C*-algebra generated by the singular Bochner-Martinelli integral over a smooth closed hypersurfaces in Cn, we compute its principal symbol. We show then that the Szegö projection belongs to the strong closure of the algebra generated by the singular Bochner-Martinelli integral.
We consider a boundary value problem for an elliptic differential operator of order 2m in a domain D ⊂ n. The boundary of D is smooth outside a finite number of conical points, and the Lopatinskii condition is fulfilled on the smooth part of δD. The corresponding spaces are weighted Sobolev spaces H(up s,Υ)(D), and this allows one to define ellipticity of weight Υ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(Υ + 2m)/n, then the root functions of the problem are complete in L²(D). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.
We describe a new algebra of boundary value problems which contains Lopatinskii elliptic as well as Toeplitz type conditions. These latter are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. Every elliptic operator is proved to admit up to a stabilisation elliptic conditions of such a kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. The crucial novelty consists of the new type of weighted Sobolev spaces which serve as domains of pseudodifferential operators and which fit well to the nature of operators.
In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.
The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah–Patodi–Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.
Contents: 1 The main notations and definitions. 2 Statement of the problems and main results. 2.1 The case of a Z-configuration. 2.2 The case of a P-configuration. 3 Proofs of Theorems 1-7. 4 Applications. 4.1 Multiplicative Cauchy type functional equation. 4.2 On some integral equations relating to a geometric problem 4.3 On the solvability of boundary problem for hyperbolic differential equations.