Fine and quasi connectedness in nonlinear potential theory

Adams, David R.; Lewis, John L.

Annales de l'Institut Fourier, Tome 35 (1985), p. 57-73 / Harvested from Numdam

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Abstract

Si $B_{α, p}$ désigne la capacité de Bessel des sous-ensembles de l’espace euclicien de dimension $n$ , $α > 0$ , $1 < p < \infty$ , associé naturellement avec l’espace des potentiels de Bessel des fonctions $L^{p}$ -functions, alors notre résultat principal est l’estimation suivante : pour $1 < α p \leq n$ , il existe une constante $C = C (α, p, n)$ de telle sorte que pour n’importe quel ensemble $E$ ,

min {B_{α, p} (E \cap Q), B_{α, p} (E^{c} \cap Q)} \leq C \cdot B_{α, p} (Q \cap \partial_{f} E)

pour tous les cubes ouverts $Q$ dans l’espace de dimension $n$ . Ici, $\partial_{f} E$ est le bord de l’ensemble $E$ dans la topologie —fine $(α, p)$ — c’est-à-dire la topologie minimale sur l’espace de dimension $n$ qui rend continu les potentiels $(α, p)$ -non-linéaires associés. Par conséquent, nous déduisons que pour $α p > 1$ , les ensembles ouverts et connexes sont connexes dans la $(α, p)$ -quasi-topologie (c’est-à-dire la topologie engendrée par la fonction de l’ensemble $B_{α, p}$ au sens de Fuglede) et que les ensembles $(α, p)$ -finement ouverts $(α, p)$ -finement connexes sont connexes par arcs. Nos méthodes sont basées sur les propriétés de Kellog-Choquet des capacités $B_{α, p}$ et certains aspects de la théorie de la mesure géométrique. Le cas newtonien classique correspond au cas $α = 1$ , $p = 2$ et $n = 3$ .

If $B_{α, p}$ denotes the Bessel capacity of subsets of Euclidean $n$ -space, $α > 0$ , $1 < p < \infty$ , naturally associated with the space of Bessel potentials of $L^{p}$ -functions, then our principal result is the estimate: for $1 < α p \leq n$ , there is a constant $C = C (α, p, n)$ such that for any set $E$

min {B_{α, p} (E \cap Q), B_{α, p} (E^{c} \cap Q)} \leq C \cdot B_{α, p} (Q \cap \partial_{f} E)

for all open cubes $Q$ in $n$ -space. Here $\partial_{f} E$ is the boundary of the $E$ in the $(α, p)$ -fine topology i.e. the smallest topology on $c$ -space that makes the associated $(α, p)$ -linear potentials continuous there. As a consequence, we deduce that for $α p > 1$ , open connected sets are connected in the $(α, p)$ -quasi topology (i.e. the topology generated by the set function $B_{α, p}$ in the sense of Fuglede), and the $(α, p)$ -finely open $(α, p)$ -finely connected sets are arcwise connected. Our methods rely on the Kellog-Choquet properties of the capacities $B_{α, p}$ and aspects of geometric measure theory. The classical Newtonian case corresponds to the case $α = 1$ , $p = 2$ and $n = 3$ .

Publié le : 1985-01-01

MR 86h:31009 | Zbl 0545.31012 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.998

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     author = {Adams, David R. and Lewis, John L.},
     title = {Fine and quasi connectedness in nonlinear potential theory},
     journal = {Annales de l'Institut Fourier},
     volume = {35},
     year = {1985},
     pages = {57-73},
     doi = {10.5802/aif.998},
     mrnumber = {86h:31009},
     zbl = {0545.31012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1985__35_1_57_0}
}

Adams, David R.; Lewis, John L. Fine and quasi connectedness in nonlinear potential theory. Annales de l'Institut Fourier, Tome 35 (1985) pp. 57-73. doi : 10.5802/aif.998. http://gdmltest.u-ga.fr/item/AIF_1985__35_1_57_0/

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