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Korevaar-Schoen $p$-energies and their $Γ$-limits on Cheeger spaces

Patricia Alonso Ruiz, Fabrice Baudoin

Abstract

This paper studies properties of $Γ$-limits of Korevaar-Schoen $p$-energies on a Cheeger space. When $p>1$, this kind of limit provides a natural $p$-energy form that can be used to define a $p$-Laplacian, and whose domain is the Newtonian Sobolev space $N^{1,p}$. When $p=1$, the limit can be interpreted as a total variation functional whose domain is the space of BV functions. When the underlying space is compact, the $Γ$-convergence of the $p$-energies is improved to Mosco convergence for every $p \ge 1$.

Korevaar-Schoen $p$-energies and their $Γ$-limits on Cheeger spaces

Abstract

This paper studies properties of -limits of Korevaar-Schoen -energies on a Cheeger space. When , this kind of limit provides a natural -energy form that can be used to define a -Laplacian, and whose domain is the Newtonian Sobolev space . When , the limit can be interpreted as a total variation functional whose domain is the space of BV functions. When the underlying space is compact, the -convergence of the -energies is improved to Mosco convergence for every .
Paper Structure (18 sections, 28 theorems, 150 equations)

This paper contains 18 sections, 28 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Definitions and setup
  3. Cheeger spaces
  4. $\Gamma$ and $\overline{\Gamma}$-convergence
  5. Useful first observations
  6. Korevaar-Schoen $p$-energy forms
  7. Existence of $\Gamma$-limits
  8. Some properties of the $p$-energy form
  9. $p$-Laplacian
  10. Korevaar-Schoen $p$-energy measures
  11. Construction of the $p$-energy measures
  12. Properties of $p$-energy measures
  13. $(p,p)$-Poincaré inequality with respect to the $p$-energy measure
  14. Absolute continuity for $p>1$
  15. Equivalence of $\Gamma_1$ with Miranda's BV measures when $p=1$
  16. ...and 3 more sections

Key Result

Proposition 2.12

For any $\varepsilon>0$ and any $f\in L^p(X,\mu)$, the function $f_\varepsilon$ in E:def_Lip_approx is locally Lipschitz with and $f_{\varepsilon}$ converges to $f$ in $L^p(X,\mu)$ as $\varepsilon\to 0^+$.

Theorems & Definitions (72)

  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8: Theorem 8.5 DMas93
  • Definition 2.9
  • Remark 2.10
  • Remark 2.11
  • Proposition 2.12
  • ...and 62 more