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R. Salimov

Publications and source records attributed to R. Salimov.

11 recordsLinked to original sources
arXiv2025

Generalized Cauchy-Riemann equations and relevant PDE

Here we give a survey of consequences from the theory of the Beltrami equations in the complex plane $\mathbb C$ to generalized Cauchy-Riemann equations $\nabla v = B \nabla u$ in the real plane $\mathbb R^2$ and clarify the relationships of the latter to the $A-$harmonic equation ${\rm div} A\,{\rm grad}\, u = 0$ with matrix valued coefficients $A$ that is one of the main equations of the potential theory, namely, of the hydro\-mechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on existence, representation and regularity of their solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincare and Riemann.

math.AP↗
arXiv2019

On Hölder continuity of mappings in domains and on boundaries

We study mappings with branching of a domain of Euclidean space. The Hölder and Lipschitz continuity are established for one class of spatial mappings whose characteristic satisfies the Dini type condition in a given domain. In addition, we found conditions on the complex coefficient of the degenerate Beltrami equations in the unit disk under which generalized homeomorphic solutions of this equation are Hölder continuous at the points of the boundary.

math.CV↗
arXiv2016

The estimation of the area of a disk image for Sobolev classes

For regular homeomorphisms of Sobolev class $W^{1,1}_{\textrm{loc}}$ having the Luzin $N$-property, it is established the estimation of the area of a disk image in terms of an angular dilatation. As a corollary, the analog of the well-known Ikoma-Schwartz lemma for such mappings is obtained.

math.CV↗
arXiv2014

On $p$-modulus estimates in Orlicz-Sobolev classes

Under a condition of the Calderon type on $φ$, we show that a homeomorphism $f$ of finite distortion in $W^{1,φ}_{\rm loc}$ and, in particular, $f\in W^{1,q}_{\rm loc}$ for $q>n-1$ in ${\Bbb R}^n$, $n\ge 3$, is a lower $Q$-homeomorphisms with respect to the $p$-modulus with $\left[ K_{I,α}(x,f)\right]^β$, $p>n-1$ and a ring $Q_{*}$-homeomorphism with respect to the $\frac{p}{p-n+1}$-modulus with $Q_{*}(x)=K_{I,α}(x,f)$ where $ K_{I,α}(x,f)$ is its inner $α$-dilatation and $α=\frac{p}{p-n+1}$, $β=\frac{p-n+1}{n-1}$.

math.CV↗
arXiv2013

Lower $Q$-Homeomorphisms With Respect To $P$-Modulus And Orlicz-Sobolev Classes

We show that under a condition of the Calderon type on $φ$ the homeomorphisms $f$ with finite distortion in $W^{1,φ}_{\rm loc}$ and, in particular, $f\in W^{1,s}_{\rm loc}$ for $s>n-1$ are the so-called lower $Q$-homeomorphisms with respect to $p$-modulus where $Q(x)$ is equal to its outer $p$-dilatation $K_{p,f}(x)$.

math.CV↗
arXiv2010

On equicontinuity of homeomorphisms with finite distortion in the plane

It is stated equicontinuity and normality of families $\frak{F}^Φ$ of the so--called homeomorphisms with finite distortion on conditions that $K_{f}(z)$ has finite mean oscillation, singularities of logarithmic type or integral constraints of the type $\intΦ\left(K_{f}(z)\right)dx\,dy<\infty$ in a domain $D\subset{\C}.$ It is shown that the found conditions on the function $Φ$ are not only sufficient but also necessary for equicontinuity and normality of such families of mappings.

math.CV↗
arXiv2010

$ACL$ and differentiability of the Open Discrete Ring (p,Q)-Mappings

We study the so--called ring $(p,Q)$--mappings which are the natural generalization of quasiregular and quasi isometric mappings. It is proved that open discrete ring $(p,Q)$--mappings are differentiable a.e. and belong to the class $ACL$ in ${\Bbb R}^n$, $n\ge 2$, furthermore, $f\in W_{loc}^{1,1}$ provided that $Q\in L^{1}_{loc}$ and $p>n-1$

math.CV↗