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Regularity results to the class of variational obstacle problems with variable exponent

Debraj Kar

Abstract

In this paper we prove local gradient estimates and higher differentiability result for the solutions of variational obstacle inequalities \int_Ω\big<\mathcal{A}(x,u,Du),D(φ-u)\big>dx\geq \int_Ω\mathcal{B}(x,u,Du)(φ-u)dx. for all $φ\in \mathcal{K}_ψ(Ω)$. Here $Ω(\subset\mathbb{R}^n)$ is bounded, $n\geq 2$ and $ψ:Ω\rightarrow\mathbb{R}$ is called obstacle. Here we deal with variable exponent growth , namely $p(.)$-growth . At first we prove Calderón-Zygmund estiamte and then using this result to prove higher differentiability result in Besov scale.

Regularity results to the class of variational obstacle problems with variable exponent

Abstract

In this paper we prove local gradient estimates and higher differentiability result for the solutions of variational obstacle inequalities \int_Ω\big<\mathcal{A}(x,u,Du),D(φ-u)\big>dx\geq \int_Ω\mathcal{B}(x,u,Du)(φ-u)dx. for all . Here is bounded, and is called obstacle. Here we deal with variable exponent growth , namely -growth . At first we prove Calderón-Zygmund estiamte and then using this result to prove higher differentiability result in Besov scale.
Paper Structure (12 sections, 22 theorems, 363 equations)

This paper contains 12 sections, 22 theorems, 363 equations.

Table of Contents

  1. Introduction
  2. Notations and statements of Main results
  3. Notations, Some Spaces and Preliminery results
  4. Space with variable exponent
  5. Difference quotient
  6. Spaces $L^p\log^\sigma L$
  7. Besov-Lipscchitz spaces
  8. Morrey Space
  9. Local Gradient Estimates
  10. Comparison estimates by approximation
  11. Proof of Theorem \ref{['T2.1']}
  12. Higher Differentiability

Key Result

Theorem 2.1

Let $u\in \mathcal{K}_\psi(\Omega)$ be a weak solution satisfying variational inequalities (eq1.3). Let assumptions $(eqh1),(eqa4),(eqa6)$ and $p(.)$ and $\mathcal{A}$ satisfies (eq2.4) and (eq2.6) respectively for sufficiently small $\delta=\delta(n,\nu',L',\gamma_1,\gamma_2,q)$. Moreover we assume . for $c=c(n,\nu',L',\gamma_1,\gamma_2,q)$

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Definition 3.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.2
  • ...and 17 more