Variation of the Liouville measure of a hyperbolic surface

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic foliation.Comment: AmsLaTeX with package epsfig, 27 pages, 3 figures; one argument corrected in Section 7, minor improvements elsewhere; to appear in Erg. Th. Dyn. Sys

Finite Difference Method for the Reverse Parabolic Problem

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example

A Note on Parabolic Differential Equations on Manifold

4th International Conference of Mathematical Sciences (ICMS) -- JUN 17-21, 2020 -- Maltepe Univ, ELECTR NETWORKThe present extended abstract considers the differential equations on smooth closed manifolds, investigates and establishes the well-posedness of nonlocal boundary value problems (NBVP) in Holder spaces. It also establishes new coercivity estimates in various Holder norms for the solutions of such boundary value problems for parabolic equations.WOS:0006642014000712-s2.0-8510224312

A Note on Hyperbolic Differential Equations on Manifold

5th International Conference on Analysis and Applied Mathematics (ICAAM) -- SEP 23-30, 2020 -- Mersin, TURKEYIn this extended abstract, considering the differential equations on hyperbolic plane. we investigate and establish the well-posedness of boundary value problem for hyperbolic equations in Holder spaces. Furthermore, we establish new coercivity estimates in various Holder norms for the solutions of such boundary value problems for hyperbolic equations.WOS:0006537346000582-s2.0-8510160595

The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

Scopu

A Note on the Parabolic Differential and Difference Equations

The differential equation u'(t)+Au(t)=f(t)(−∞<t<∞) in a general Banach space E with the strongly positive operator A is ill-posed in the Banach space C(E)=C(ℝ,E) with norm ‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder space Cα(E)=Cα(ℝ,E) with norm ‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα), 0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in C(ℝτ,E) spaces is proved. The well-posedness of this difference scheme in Cα(ℝτ,E) spaces is obtained