Piecewise Holder Research Articles

Regularity Results for Viscous 3D Boussinesq Temperature Fronts

This paper is about the dynamics of non-diffusive temperature fronts evolving by the incompressible viscous Boussinesq system in $${\mathbb {R}}^3$$ . We provide local in time existence results for initial data of arbitrary size. Furthermore, we show global in time propagation of regularity for small initial data in critical spaces. The developed techniques allow to consider general fronts where the temperature is piecewise Holder (not necessarily constant), which preserve their structure together with the regularity of the evolving interface.

On the Uniqueness of a Solution to an Inverse Problem of Scattering by an Inhomogeneous Solid with a Piecewise Hölder Refractive Index in a Special Function Class

The problem of reconstructing a piecewise Holder continuous function describing the refractive index of an inhomogeneous obstacle scattering a monochromatic wave is considered. The boundary value scattering problem is reduced to a system of integral equations. The equivalence of the integral and differential formulations of the problem is proved. A two-step method for solving the inverse problem is proposed. A linear integral equation of the first kind is solved at the first step. Sufficient conditions for the uniqueness of its solution in the class of piecewise constant functions are obtained. At the second step, the unknown refractive index is explicitly expressed in terms of the solution obtained at the first step.

Dynamical Bounds for Quasiperiodic Schrödinger Operators with Rough Potentials

We establish localization type dynamical bounds as a corollary of positive Lyapunov exponents for general operators with quasiperiodic potentials defined by piecewise Holder functions.

On an Elliptic Equation Arising from Composite Materials

In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation \begin{equation*} D_i(a(x)D_iu)=D_if_i \end{equation*} in $\mathbb{R}^2$, where $a(x)$ and $f_i(x)$ are piecewise H\"older continuous in a domain containing two touching balls as subdomains. When $f_i\equiv 0$ and $a$ is piecewise constant, we prove that $u$ is piecewise smooth with bounded derivatives. This completely answers a question raised by Li and Vogelius [7] in dimension 2.

On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Holder coefficients in a Holder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Holder coefficients in a Holder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.