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Potential analysis of multi layers optic fiber models

Kateryna Buryachenko, Yuliya Kudrych

TL;DR

This work develops pointwise Wolff-potential estimates for nonnegative weak solutions of double-phase elliptic equations with variable exponents $p(x)$ and $q(x)$, motivated by modeling multilayer optic-fiber devices. Building on prior constant-exponent results, it introduces the $W^{1,G}$ framework and derives sharp pointwise bounds for $u(x_0)$ in terms of nonlinear Wolff potentials of the right-hand side $f\in L^1$, with distinct cases depending on the interlayer coefficient $a(x_0)$. The main theorem extends known regularity and potential-estimate techniques to the variable-exponent setting and provides explicit dependence on the layer structure via $p_-,p_+,q_-,q_+$. These results offer a rigorous mathematical tool for analyzing heterogeneous optical-media models and can inform design and analysis of multilayer optic-fiber technologies.

Abstract

We study pointwise potential estimates of the weak nonnegative solutions for the double phase elliptic equations with variables powers of nonlinearity: p(x), q(x). We discuss also the applications of the obtained theoretical results for the problem of modeling and analyzing of optic fiber and optic cable modern technologies.

Potential analysis of multi layers optic fiber models

TL;DR

This work develops pointwise Wolff-potential estimates for nonnegative weak solutions of double-phase elliptic equations with variable exponents and , motivated by modeling multilayer optic-fiber devices. Building on prior constant-exponent results, it introduces the framework and derives sharp pointwise bounds for in terms of nonlinear Wolff potentials of the right-hand side , with distinct cases depending on the interlayer coefficient . The main theorem extends known regularity and potential-estimate techniques to the variable-exponent setting and provides explicit dependence on the layer structure via . These results offer a rigorous mathematical tool for analyzing heterogeneous optical-media models and can inform design and analysis of multilayer optic-fiber technologies.

Abstract

We study pointwise potential estimates of the weak nonnegative solutions for the double phase elliptic equations with variables powers of nonlinearity: p(x), q(x). We discuss also the applications of the obtained theoretical results for the problem of modeling and analyzing of optic fiber and optic cable modern technologies.

Paper Structure

This paper contains 4 sections, 1 theorem, 19 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Main result
  4. Applications to the multi layers optic fiber models

Key Result

Theorem 3.1

Let $u\in W^{1,G}(\Omega)\cap L^{\infty}$ be a nonnegative weak solution to Eq. (eq2.0). Let conditions (cond_powers) be satisfied and let $[a]_{C^{0,\alpha}(\Omega)}:=\underset{x,y\in\Omega,\,x\neq y}{\rm sup}\frac{|a(x)-a(y)|}{|x-y|^{\alpha}}.$ Assume also that the point $x_0\in\Omega$ is such tha If $a(x_0)>0$ and $\rho_0^{\alpha}=\frac{a(x_0)}{4 [a]_{C^{0,\alpha}(\Omega)}}\geq\rho^{\alpha},$ t

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Remark 3.1