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 , p. 151-182
Zbl 0488.35034 | MR 84g:35067 | 4 citations dans Numdam
doi : 10.5802/aif.883
URL stable : http://www.numdam.org/item?id=AIF_1982__32_3_151_0

Nous considérons des équations elliptiques dégénérées, de la forme i,j D x i (a ij (x)D x j ),λw(x)|ξ| 2 i,j a ij (x)ξ i ξ j Λw(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 i,j D x i (a ij (x)D x j ),whereλw(x)|ξ| 2 i,j a ij (x)ξ i ξ j Λw(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.

Bibliographie

[1] L. Carleson, Selected Problems on Exceptional Sets, 1967, Van Nostrand. MR 37 #1576 | Zbl 0189.10903

[2] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. MR 50 #10670 | Zbl 0291.44007

[3] J. Deny, Théorie de la capacité dans les espaces fonctionnels, Séminaire Brelot-Choquet-Deny (Théorie du Potentiel). no. 1, (1964-1965), 1-13. Numdam | Zbl 0138.36605

[4] E.B. Fabes, D.S. Jerison and C.E. Kenig, Boundary behavior of solutions of degenerate elliptic equations, preprint.

[5] E.B. Fabes, C.E. Kenig, and R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. in P.D.E., 7(1) (1982), 77-116. MR 84i:35070 | Zbl 0498.35042

[6] F. Gehring, The Lp integrability of the partial derivatives of a quasi conformal mapping, Acta Math., 130 (1973), 266-277. MR 53 #5861 | Zbl 0258.30021

[7] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, 1980, Academic Press, N.Y., N.Y. MR 81g:49013 | Zbl 0457.35001

[8] W. Littman, G. Stampacchia and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. della Scuola Normale Sup. di Pisa, S. 3, vol. 17 (1963), 45-79. Numdam | MR 28 #4228 | Zbl 0116.30302