Dynamic problem of a power-law graded half-plane and an associated Carleman problem for two functions

Y. A. Antipov

Home/Archive/arXiv

arXiv · 2407.04148

Dynamic problem of a power-law graded half-plane and an associated Carleman problem for two functions

Y. A. Antipov

View original source ↗Read source document ↗Export metadata ↓

Abstract

A steady state plane problem of an inhomogeneous half-plane subjected to a load running along the boundary at subsonic speed is analyzed. The Lame coefficients and the density of the half-plane are assumed to be power functions of depth. The model is different from the standard static model have been used in contact mechanics since the Sixties and originated from the 1964 Rostovtsev exact solution of the Flamant problem of a power-law graded half-plane. To solve the governing dynamic equations with variable coefficients written in terms of the displacements, we propose a method that, by means of the Fourier and Mellin transforms, maps the model problem to a Carleman boundary value problem for two meromorphic functions in a strip with two shifts or, equivalently, to a system of two difference equations of the second order with variable coefficients. By partial factorization the Carleman problem is recast as a system of four singular integral equations on a segment with a fixed singularity and highly oscillating coefficients. A numerical method for its solution is proposed and tested. Numerical results for the displacement and stress fields are presented and discussed.

Explore related subjects

math.CV math-ph math.MP

Keep this discovery

BibTeX RIS

Y. A. Antipov. 2024-07-04. Dynamic problem of a power-law graded half-plane and an associated Carleman problem for two functions. https://arxiv.org/abs/2407.04148

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The two-dimensional Matkowski--Sut\^o equation with holomorphic and strictly increasing generators

We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.

math.CV↗

arXiv2026

A counterexample to an open problem of Dorff

The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized convex harmonic mapping with bounded image must remain in the same class. We construct a normalized sense-preserving harmonic diffeomorphism that maps the unit disk onto an ellipse; its self-convolution has vanishing Jacobian at some interior point of the unit disk, which provides a negative answer to Dorff's open problem.

math.CV↗

arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a K\"ahler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗