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On solvability in the small of higher order elliptic equations in Orlicz-Sobolev spaces

Javad A. Asadzade

Abstract

In this article, we consider a higher-order elliptic equation with nonsmooth coefficients with respect to Orlicz spaces on the domain $Ω\subset\mathbb{R}^{n}$. The separable subspace of this space is distinguished in which infinitely differentiable and compactly supported functions are dense; Sobolev spaces generated by these subspaces are determined. We demonstrate the local solvability of the equation in Orlicz-Sobolev spaces under specific restrictions on the coefficients of the equation and the Boyd indices of the Orlicz space. This result strengthens the previously known classical $L_{p}$ analog.

On solvability in the small of higher order elliptic equations in Orlicz-Sobolev spaces

Abstract

In this article, we consider a higher-order elliptic equation with nonsmooth coefficients with respect to Orlicz spaces on the domain . The separable subspace of this space is distinguished in which infinitely differentiable and compactly supported functions are dense; Sobolev spaces generated by these subspaces are determined. We demonstrate the local solvability of the equation in Orlicz-Sobolev spaces under specific restrictions on the coefficients of the equation and the Boyd indices of the Orlicz space. This result strengthens the previously known classical analog.
Paper Structure (8 sections, 14 theorems, 104 equations)

This paper contains 8 sections, 14 theorems, 104 equations.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Elliptic operator of mth order
  4. Orlicz spaces
  5. Some important results
  6. The Orlicz-Sobolev space $W^{m}_{G_{M},d_{\Omega}}(\Omega)$. Main lemma
  7. Local existence theorem
  8. Conclusion

Key Result

Theorem 2.1

(2) For any mth order elliptic operator $L_{0}$ of the form (2.3) with the constant coefficients, the function $\mathbb{J}(x)$ can be constructed which has the below properties: 1). If n is odd or n is even and $n>m$, hence where $\omega(x)$ is a positive homogeneous function of degree zero ($\omega(tx)=\omega(x),\quad \forall t>0$) If $n$ is even and $n\leq m$, hence where $q$ is homogeneous po

Theorems & Definitions (23)

  • Theorem 2.1
  • Lemma 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.1
  • Theorem 2.4
  • Theorem 2.5
  • ...and 13 more