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

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

min{Bα,p(EQ),Bα,p(EcQ)}C·Bα,p(QfE)

for all open cubes Q in n-space. Here 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 αp>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.

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

min{Bα,p(EQ),Bα,p(EcQ)}C·Bα,p(QfE)

pour tous les cubes ouverts Q dans l’espace de dimension n. Ici, 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 αp>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.

@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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {35},
     number = {1},
     year = {1985},
     pages = {57-73},
     doi = {10.5802/aif.998},
     zbl = {0545.31012},
     mrnumber = {86h:31009},
     language = {en},
     url = {http://www.numdam.org/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, Volume 35 (1985) no. 1, pp. 57-73. doi : 10.5802/aif.998. http://www.numdam.org/item/AIF_1985__35_1_57_0/

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