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Approximation of $L^\infty$ functionals with generalized Orlicz norms

Giacomo Bertazzoni, Michela Eleuteri, Elvira Zappale

Abstract

The aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. $Γ$-convergence results and related representation theorems in terms of $L^\infty$ functionals are proven for sequences of generalized Orlicz energies under mild convexity assumptions. This latter hypothesis is removed in the variable exponent setting.

Approximation of $L^\infty$ functionals with generalized Orlicz norms

Abstract

The aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. -convergence results and related representation theorems in terms of functionals are proven for sequences of generalized Orlicz energies under mild convexity assumptions. This latter hypothesis is removed in the variable exponent setting.

Paper Structure

This paper contains 12 sections, 13 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Generalized Orlicz spaces
  3. Some technical lemmas and crucial examples
  4. Variable exponents Lebesgue-Sobolev spaces
  5. Preliminary results
  6. Young measures
  7. Convexity notions for supremal functionals
  8. $\Gamma-$convergence
  9. Lower growth approximation
  10. Statement of the main results
  11. Proofs of Theorems
  12. $\mathbf{L}^{p(\cdot)}$- approximation of supremal functionals under no convexity assumptions

Key Result

Lemma 2.6

[Unit ball property, see HarH19] If $\phi \in \Phi_w(\Omega)$, then

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 21 more