The Wiener test for degenerate elliptic equations

Fabes, E. B.; Jerison, D. S.; Kenig, C. E.

doi:10.5802/aif.883

Fabes, E. B. ; Jerison, D. S. ; Kenig, C. E.

Annales de l'Institut Fourier, Tome 32 (1982) no. 3, pp. 151-182.

Résumé
Abstract

Nous considérons des équations elliptiques dégénérées, de la forme

\sum_{i, j} D_{x_{i}} (a_{i j} (x) D_{x_{j}}), où λ w (x) | ξ |^{2} \leq \sum_{i, j} a_{i j} (x) ξ_{i} ξ_{j} \leq Λ w (x) | ξ |^{2} .

En faisant des hypothèses convenables sur $w$ , nous obtenons une caractérisation du type de Weiner, (utilisant des capacités avec poids), pour l’ensemble de points réguliers de ces opérateurs. Nous montrons que l’ensemble de points réguliers dépend seulement de $w$ . L’outil fondamental que nous utilisons est une estimation pour la fonction de Green, par rapport à $w$ .

We consider degenerated elliptic equations of the form

\sum_{i, j} D_{x_{i}} (a_{i j} (x) D_{x_{j}}), where λ w (x) | ξ |^{2} \leq \sum_{i, j} a_{i j} (x) ξ_{i} ξ_{j} \leq Λ w (x) | ξ |^{2} .

Under suitable assumptions on $w$ , we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$ . The main tool we use is an estimate for the Green function in terms of $w$ .

MR Zbl | 6 citations dans Numdam

DOI : 10.5802/aif.883

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     author = {Fabes, E. B. and Jerison, D. S. and Kenig, C. E.},
     title = {The {Wiener} test for degenerate elliptic equations},
     journal = {Annales de l'Institut Fourier},
     pages = {151--182},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {32},
     number = {3},
     year = {1982},
     doi = {10.5802/aif.883},
     mrnumber = {84g:35067},
     zbl = {0488.35034},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.883/}
}

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Fabes, E. B.; Jerison, D. S.; Kenig, C. E. The Wiener test for degenerate elliptic equations. Annales de l'Institut Fourier, Tome 32 (1982) no. 3, pp. 151-182. doi : 10.5802/aif.883. https://www.numdam.org/articles/10.5802/aif.883/

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