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Convergence of generalized Orlicz norms with lower growth rate tending to infinity

Giacomo Bertazzoni, Petteri Harjulehto, Peter Hästö

Abstract

We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea--Mihăilescu (Orlicz case) and Eleuteri--Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.

Convergence of generalized Orlicz norms with lower growth rate tending to infinity

Abstract

We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea--Mihăilescu (Orlicz case) and Eleuteri--Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.
Paper Structure (4 sections, 8 theorems, 43 equations)

This paper contains 4 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Asymptotically sharp embeddings
  4. Main theorems

Key Result

Lemma 2.2

Let $f: \mathbb{R}^N \to \mathbb{R}$ be a lower semicontinuous and level convex function and let $\mu$ be a probability measure in the open set $U \subset \mathbb{R}^N$. Then for every $u \in L^1_{\mu}(U, \mathbb{R}^N)$.

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.2: Theorem 1.2, BJW
  • Definition 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 14 more