We prove that, for any finite Blaschke product w=B(z) in the unit disk, corresponding Riemann surface over w–plane contains a one-sheeted disk of radius 0.5. Moreover, it with radial slit. apply this result to obtain universal lower estimate Bloch seminorm products.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three mod...

We consider an optimal rearrangement maximization problem involving the fractional Laplace operator ( − Δ ) s , 0 < 1 and Gagliardo-Nirenberg seminorm [ u ] . prove existence of a maximizer, analyze its properties show that it satisfies unstable obstacle equation for some α > = χ { }

Abstract. Numerical solution of one-dimensional elliptic problems is investigated using an averaged discontinuous discretization. The corresponding numerical method can be performed using the favorable properties of the discontinuous Galerkin (dG) approach, while for the average an error estimation is obtained in the H-seminorm. We point out that this average can be regarded as a lower order mo...

1221 x(T) is nonzero. Based on this remark, it was erroneously claimed 1 that W n has no usual extension. From Proposition 1 one obtains the following result. Proposition 2: The norm of ET x in W n is kE T xk W = kxk 2 2;T + k _ xk 2 2;T + kx(T)k 2 1=2 (1) where k1k2;T denotes the seminorm in L n 2;e defined by k1k2;T = kP T 1k 2. Set k1k W;T := kE T 1 k W (and notice the difference with the de...

Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ Cd(R), with lower-order partials vanishing at infinity and dth-order partials vanishing as ‖x‖−(d+1+ε), ε > 0, on any domain ⊂ R with unit Lebesgue measure, the L2( )-error in approximati...

It is well known that finite element solutions for elliptic PDEs with Dirac measures as source terms converge, due to the fact that the solution is not in H1, suboptimal in classical norms. A standard remedy is to use graded meshes, then quasioptimality, i.e., optimal up to a log-factor, for low order finite elements can be recovered, e.g., in the L2-norm. Here we show for the lowest order case...

We provide necessary and sufficient conditions for a 1-jet (f,G):E→R×X to admit an extension (F,∇F) some F∈C1,ω(X). Here E stands arbitrary subset of Hilbert space X ω is modulus continuity. As corollary, in the particular case X=Rn, we obtain (nonlinear) operator whose norm does not depend on dimension n. Furthermore, construct extensions such way that: (1) (f,G)↦(F,∇F) bounded with respect na...