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Nonlocal convolution type functionals and related Orlicz spaces

Denis Borisov, Andrey Piatnitski

Abstract

In the paper we introduce Orlicz type functional spaces defined in terms of nonlocal convolution type integral functionals and study the main properties of these spaces. We show in particular that, under natural convexity and growth conditions on the integrand, the corresponding spaces are Banach and separable. We also characterize the dual spaces and provide a number of examples.

Nonlocal convolution type functionals and related Orlicz spaces

Abstract

In the paper we introduce Orlicz type functional spaces defined in terms of nonlocal convolution type integral functionals and study the main properties of these spaces. We show in particular that, under natural convexity and growth conditions on the integrand, the corresponding spaces are Banach and separable. We also characterize the dual spaces and provide a number of examples.
Paper Structure (16 sections, 18 theorems, 232 equations)

This paper contains 16 sections, 18 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Problem setup and main results
  3. Main results
  4. Discussion of main results
  5. Preliminaries
  6. Banach space and equivalent norm
  7. Density of smooth functions and separability
  8. Space $\Lambda(\Omega)$
  9. Bounded domain with Lipschitz boundary
  10. Arbitrary domain
  11. Dual spaces
  12. Auxiliary statements
  13. Dual space for $\Lambda(\Omega)$
  14. Dual space for $\mathcal{L}(\Omega)$
  15. Bounded domain
  16. ...and 1 more sections

Key Result

Theorem 2.1

Assume that Conditions Meas, Conv, Pconv, Bound are satisfied. Then the functional $\|u\|_{\mathcal{L}(\Omega)}:=f(u)$ is a norm, and the space $\mathcal{L}(\Omega)$ equipped with this norm is Banach. The following embeddings are valid The embeddings are continuous.

Theorems & Definitions (30)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Lemma 2.1
  • ...and 20 more