In the following work, we consider Boltzmann equation that models a polyatomic gas by representing microscopic internal energy continuous variable I. Under some convenient assumptions on transition function $ \mathcal{B} $, prove linearized operator \mathcal{L} of this model is Fredholm operator. For this, write as perturbation collision frequency multiplication operator, and \mathcal{K} compac...

The invertibility of second-order finite-difference operators with constant operator coefficients acting in the Banach space two-sided vector sequences is proved under condition their uniform injectivity (in particular, left invertibility), or surjectivity right Fredholm property.

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then is Fredholm. This generalizes a theorem due to B\"ar-Strohmaier in case of finite times, and we also show corresponding index formula extends setting. Furthermore, demonstrate existe...

We present a simple example for the failure of Calder\'on-Zygmund estimate $\bar{\partial}$-operator when Sobolev $(k,p)$-norms are replaced by $C^k$-norms. This is discussed in context elliptic bootstrapping, Fredholm theory, and regularity $J$-holomorphic curves.

We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform λ−adjustment which is weaker than perturbations by small gap, operator norm, q−norm, and K2−approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform λ−adjustment, while other fail. However, uniformly λ−adjusted subspaces...

In this paper we extend the results obtained in [9, 10] to manifolds with SpinC-structures defined, near the boundary, by an almost complex structure. We show that on such a manifold with a strictly pseudoconvex boundary, there are modified ∂̄-Neumann boundary conditions defined by projection operators, Reo + , which give subelliptic Fredholm problems for the SpinC-Dirac operator, ð eo + . We in...

Let L := −r−2(r∂r )2 − ∂2 z . We consider the equation Lu = f on a bounded polygonal domain with suitable boundary conditions, derived from the three-dimensional axisymmetric Poisson’s equation. We establish the well-posedness, regularity, and Fredholm results in weighted Sobolev spaces, for possible singular solutions caused by the singular coefficient of the operator L, as r → 0, and by non-s...

Let X be a Spin manifold with boundary, such that the Spin structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szego projectors, we define modified partial differential-Neumann boundary conditions, Reo, for spinors, which lead to subelliptic Fredholm boundary value problems for the S...