The goal of this paper is to obtain the regularity and the existence of solutions of a retarded semilinear differential equation with nonlocal condition by applying Schauder’s fixed point theorem. We construct the fundamental solution, establish the Hölder continuity results concerning the fundamental solution of its corresponding retarded linear equation, and prove the uniqueness of solutions ...

This article introduces a method for computing upper and lower bounds for the logarithmic capacity of a compact plane set. If the set has the Hölder continuity property, then these bounds converge to the value of the capacity. A number of examples are discussed in detail, including the Cantor middle-third set, for which we estimate c(E) ≈ 0.220949102189507.

Enstrophy is an averagedmeasure of fluid vorticity. This quantity is particularly important in rotating geophysical flows. We investigate the dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under random wind forcing. We obtain upper bounds on the enstrophy, as well as results establishing its Hölder continuity and describing the small-time asymptotics.

We prove an interpolation type inequalitie between C , L∞ and L spaces and use it to establish the local Hölder continuity of the inverse of the p-Laplace operator: ‖(−∆p)(f)−(−∆p)(g)‖C1(Ω̄) ≤ C‖f−g‖ r L(Ω) , for any f and g in a bounded set in L(Ω).

We consider Hölder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer’s Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the no...

In this paper, we consider certain affine similarity of refinable functions and establish certain connection between some local and global properties of refinable functions, such as local and global linear independence , local smoothness and B-spline, local and global Hölder continuity.

The maximum principle for SPDEs is established in multidimensional C domains. An application is given to proving the Hölder continuity up to the boundary of solutions of one-dimensional SPDEs. The maximum principle is one of the most powerful tools in the theory of second-order elliptic and parabolic partial differential equations. However, until now it did not play any significant role in the ...