In this paper we investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain – including a very weak compatibility condition between the Dirichlet boundary part and its complement – we prove Hölder continuity of the solu...

We study the system curl (a(x) curlu) = 0, divu = 0 with a bounded measurable coefficient a(x). The main result of this paper is the Hölder continuity of weak solutions of the system above. As an application, we prove the Cα regularity of weak solutions of the Maxwell’s equations in a quasi-stationary electromagnetic field.

It is shown that a composite Julia set generated by an infinite array of polynomial mappings is strongly analytic when regarded as a multifunction of the generating maps. An example of such a multifunction, the values of which have Hölder Continuity Property, is constructed. 1

The joint continuity of local times is proved for a large class of anisotropic Gaussian random fields. Sharp local and global Hölder conditions for the local times under an anisotropic metric are established. These results are useful for studying sample path and fractal properties of the Gaussian fields.

In two dimensions every weak solution to a nonlinear elliptic system div a(x, u,Du) = 0 has Hölder continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent p ≥ 2. We give an example showing that this result cannot be extended to the subquadratic case, i.e. that weak solutions are not necessarily continuous if 1 < p < 2.

This paper contains bounds for the distortion in the spherical metric, that is to say bounds for the constant of Hölder continuity of mappings f : (R, q) → (R, q) where q denotes the spherical metric. The mappings considered are K-quasiconformal (K ≥ 1) and satisfy some normalizations or restrictions. All bounds are explicit and asymptotically sharp as K → 1.

We study generalized fractional p-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions such nonlocal problems by finding a suitable Sobolev-Poincaré inquality.

Comparison results of the nonlinear scalar Riemann-Liouville fractional differential equation of order q, 0 < q ≤ 1, are presented without requiring Hölder continuity assumption. Monotone method is developed for finite systems of fractional differential equations of order q, using coupled upper and lower solutions. Existence of minimal and maximal solutions of the nonlinear fractional different...

This paper contains bounds for the distortion in the spherical metric, that is to say, bounds for the constant of Hölder continuity of mappings f : (Rn, q) → (Rn, q) where q denotes the spherical metric. The mappings considered are K-quasiconformal (K ≥ 1) and satisfy some normalizations or restrictions. All bounds are explicit and asymptotically sharp as K → 1.

We prove that the non smooth part of a non compactly supported potential q in the Schrödinger Hamiltonian −∆ + q, in dimension n = 2, is contained in its Born approximation qB for backscattering data in a precise sense in terms of continuity: Given q in W the difference q−qB is in the Hölder class Λ for any β < α.