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Relaxation for degenerate nonlinear functionals in the onedimensional case

Valeria Chiadò Piat, Virginia De Cicco, Anderson Melchor Hernandez

TL;DR

The paper studies the relaxation of a degenerate nonlinear functional in one dimension, where F(u) is the weighted p-energy with weight w. A key idea is to construct an auxiliary weight hat w_p dependent on p and w to define an appropriate domain and to enable L^p convergence with respect to (hat w_p)^{p-1}. A weighted Poincaré inequality with a double weight is proved, linking the weighted gradient energy to the weighted L^p norm and enabling density results for AC functions. Under the finitely degenerate weight assumption, the authors derive an explicit relaxation: Fbar(u) equals the weighted energy on the degeneracy set for u in Dom_w and is infinite otherwise; AC(Ω̄) density in the associated Sobolev space is established, while the non-finitely degenerate case remains open.

Abstract

In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight $w$. Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting $p$-growth for $1< p<+\infty$. We adopt the approach developed in [6], where some assumptions like doubling or Muckenhoupt conditions are dropped. Our main tools consist of proving the validity of a weighted Poincaré inequality involving an auxiliary weight.

Relaxation for degenerate nonlinear functionals in the onedimensional case

TL;DR

The paper studies the relaxation of a degenerate nonlinear functional in one dimension, where F(u) is the weighted p-energy with weight w. A key idea is to construct an auxiliary weight hat w_p dependent on p and w to define an appropriate domain and to enable L^p convergence with respect to (hat w_p)^{p-1}. A weighted Poincaré inequality with a double weight is proved, linking the weighted gradient energy to the weighted L^p norm and enabling density results for AC functions. Under the finitely degenerate weight assumption, the authors derive an explicit relaxation: Fbar(u) equals the weighted energy on the degeneracy set for u in Dom_w and is infinite otherwise; AC(Ω̄) density in the associated Sobolev space is established, while the non-finitely degenerate case remains open.

Abstract

In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight . Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting -growth for . We adopt the approach developed in [6], where some assumptions like doubling or Muckenhoupt conditions are dropped. Our main tools consist of proving the validity of a weighted Poincaré inequality involving an auxiliary weight.
Paper Structure (5 sections, 8 theorems, 90 equations, 1 figure)

This paper contains 5 sections, 8 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weighted Poincaré inequalities
  3. An auxiliary weight
  4. A Poincaré inequality with a double weight
  5. Relaxation for finitely degenerate weights

Key Result

Lemma 2.3

Let $(u_h)_h\subset AC([a,b])$ such that Then there exists a subsequence $(u_{h_k})$ and a function $u:\,I_{\Omega,w}\to{\mathbb{R}}$ such that

Figures (1)

  • Figure 1: In the first figure on the left hand side, we have the profile of $w(x)=(1-x^{2})^{2}$ for $x\in (-2,2)$, while in the right hand side, we have its associated weight $\hat{w}_{2}$. In this case, we note that $N_{w}=3$.

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Fundamental convergence
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • Corollary 2.9
  • Theorem 2.10: Poincaré type inequality on ${\rm{Dom}}_w$
  • ...and 5 more