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Let Hsub(0), Hsub(1) be Hilbert spaces and L : Hsub(0) -> Hsub(1) be a linear bounded operator with ||L|| ≤ 1. Then L*L is a bounded linear self-adjoint non-negative operator in the Hilbert space Hsub(0) and one can use the Neumann series ∑∞sub(v=0)(I - L*L)v L*f in order to study solvability of the operator equation Lu = f. In particular, applying this method to the ill-posed Cauchy problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smooth coefficients we obtain solvability conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauchy-Riemann system in C the summands of the Neumann series are iterations of the Cauchy type integral. We also obtain similar results 1) for the equation Pu = f in Sobolev spaces, 2) for the Dirichlet problem and 3) for the Neumann problem related to operator P*P if P is a homogeneous first order operator and its coefficients are constant. In these cases the representations involve sums of series whose terms are iterations of integro-differential operators, while the solvability conditions consist of convergence of the series together with trivial necessary conditions.
We construct a deformation quantization on an infinite-dimensional symplectic space of regular connections on an SU(2)-bundle over a Riemannian surface of genus g ≥ 2. The construction is based on the normal form thoerem representing the space of connections as a fibration over a finite-dimensional moduli space of flat connections whose fibre is a cotangent bundle of the infinite-dimensional gauge group. We study the reduction with respect to the gauge groupe both for classical and quantum cases and show that our quantization commutes with reduction.
The paper is devoted to pseudodifferential boundary value problems in domains with cuspidal wedges. Concerning the geometry we even admit a more general behaviour, namely oscillating cuspidal wedges. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to edges.
Generalizing an idea of I. Vekua [1] who, in order to construct theory of plates and shells, fields of displacements, strains and stresses of threedimensional theory of linear elasticity expands into the orthogonal Fourier-series by Legendre Polynomials with respect to the variable along thickness, and then leaves only first N + 1, N = 0, 1, ..., terms, in the bar model under consideration all above quantities have been expanded into orthogonal double Fourier-series by Legendre Polynomials with respect to the variables along thickness, and width of the bar, and then first (Nsub(3) + 1)(Nsub(2) + 1), Nsub(3), Nsub(2) = 0, 1,..., terms have been left. This case will be called (Nsub(3), Nsub(2)) approximation. Both in general (Nsub(3), Nsub(2)) and in particular (0,0) (1,0) cases of approximation, the question of wellposedness of initial and boundary value problems, existence and uniqueness of solutions have been investigated. The cases when variable cross-section turns into segments of straight line, and points have been also considered. Such bars will be called cusped bars (see also [2]).
The paper deals with a non-linear singular partial differential equation: (E) t∂/∂t = F(t, x, u, ∂u/∂x) in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by Gérard-Tahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case.