# ANNALES DE L'INSTITUT FOURIER

Fine and quasi connectedness in nonlinear potential theory
Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 57-73.

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

 $min\left\{{B}_{\alpha ,p}\left(E\cap Q\right),{B}_{\alpha ,p}\left({E}^{c}\cap Q\right)\right\}\le C·{B}_{\alpha ,p}\left(Q\cap {\partial }_{f}E\right)$

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

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

 $min\left\{{B}_{\alpha ,p}\left(E\cap Q\right),{B}_{\alpha ,p}\left({E}^{c}\cap Q\right)\right\}\le C·{B}_{\alpha ,p}\left(Q\cap {\partial }_{f}E\right)$

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

@article{AIF_1985__35_1_57_0,
author = {Adams, David R. and Lewis, John L.},
title = {Fine and quasi connectedness in nonlinear potential theory},
journal = {Annales de l'Institut Fourier},
pages = {57--73},
publisher = {Imprimerie Louis-Jean},
volume = {35},
number = {1},
year = {1985},
doi = {10.5802/aif.998},
mrnumber = {86h:31009},
zbl = {0545.31012},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.998/}
}
Adams, David R.; Lewis, John L. Fine and quasi connectedness in nonlinear potential theory. Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 57-73. doi : 10.5802/aif.998. https://aif.centre-mersenne.org/articles/10.5802/aif.998/

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