MSC 2010: 45DB05, 45E05, 78A45
We show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation.
Mathematics Subject Classification: 26D10.
The sharp constant is obtained for the Hardy-Stein-Weiss inequality for
fractional Riesz potential operator in the space L^p(R^n, ρ) with the power
weight ρ = |x|^β. As a corollary, the sharp constant is found for a similar
weighted inequality for fractional powers of the Beltrami-Laplace operator
on the unit sphere.
Mathematics Subject Classification 2010: 45DB05, 45E05, 78A45.
We give a procedure to reduce a hypersingular integral equation, arising
in 2d diffraction problems on cracks in elastic media, to a Fredholm integral
equation of the second kind, to which it is easier and more effectively to
apply numerical methods than to the initial hypersingular equation.
2000 Mathematics Subject Classification: 26A33, 42B20
There is given a generalization of the Marchaud formula for one-dimensional
fractional derivatives on an interval (a, b), −∞ < a < b ≤ ∞, to the
multidimensional case of functions defined on a region in R^n
MSC 2010: 26A33
We study mixed Riemann-Liouville integrals of functions of two variables in Hölder spaces of different orders in each variables. We consider Hölder spaces defined both by first order differences in each variable and also by the mixed second order difference, the main interest being in the evaluation of the latter for the mixed fractional integral in both the cases where the density of the integral belongs to the Hölder class defined by usual or mixed differences. The...
2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30
We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain class of one-dimensional integral operators M with a weak singularity in the kernel, from the variable exponent Lebesgue space L^p(·) ([a, b], ?) to the
Sobolev type space L^α,p(·) ([a, b], ?) of fractional smoothness. We also give formulas of closed form solutions ϕ ∈ L^p(·)
of the 1st kind integral equation M0ϕ = f, known as...
Mathematics Subject Classification: 26D10, 46E30, 47B38
We prove the Hardy inequality and a similar inequality for the dual Hardy operator for variable exponent Lebesgue spaces.
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