# Fine and quasi connectedness in nonlinear potential theory

Annales de l'institut Fourier (1985)

- Volume: 35, Issue: 1, page 57-73
- ISSN: 0373-0956

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topAdams, David R., and Lewis, John L.. "Fine and quasi connectedness in nonlinear potential theory." Annales de l'institut Fourier 35.1 (1985): 57-73. <http://eudml.org/doc/74668>.

@article{Adams1985,

abstract = {If $B_\{\alpha ,p\}$ denotes the Bessel capacity of subsets of Euclidean $n$-space, $\alpha >0$, $1< p< \infty $, naturally associated with the space of Bessel potentials of $L^p$-functions, then our principal result is the estimate: for $1< \alpha p\le n$, there is a constant $C = C(\alpha ,p,n)$ such that for any set $E$\begin\{\} \min \lbrace B\_\{\alpha ,p\}(E\cap Q),B\_\{\alpha ,p\}(E^ c\cap Q)\rbrace \le C\cdot B\_\{\alpha ,p\}(Q\cap \partial \_ fE) \end\{\}for all open cubes $Q$ in $n$-space. Here $\partial _f E$ is the boundary of the $E$ in the $(\alpha ,p)$-fine topology i.e. the smallest topology on $c$-space that makes the associated $(\alpha ,p)$-linear potentials continuous there. As a consequence, we deduce that for $\alpha p>1$, open connected sets are connected in the $(\alpha ,p)$-quasi topology (i.e. the topology generated by the set function $B_\{\alpha ,p\}$ in the sense of Fuglede), and the $(\alpha ,p)$-finely open $(\alpha ,p)$-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 $n=3$.},

author = {Adams, David R., Lewis, John L.},

journal = {Annales de l'institut Fourier},

keywords = {nonlinear potential theory; fine topologies; quasi topologies; Bessel capacities; method of balayage},

language = {eng},

number = {1},

pages = {57-73},

publisher = {Association des Annales de l'Institut Fourier},

title = {Fine and quasi connectedness in nonlinear potential theory},

url = {http://eudml.org/doc/74668},

volume = {35},

year = {1985},

}

TY - JOUR

AU - Adams, David R.

AU - Lewis, John L.

TI - Fine and quasi connectedness in nonlinear potential theory

JO - Annales de l'institut Fourier

PY - 1985

PB - Association des Annales de l'Institut Fourier

VL - 35

IS - 1

SP - 57

EP - 73

AB - If $B_{\alpha ,p}$ denotes the Bessel capacity of subsets of Euclidean $n$-space, $\alpha >0$, $1< p< \infty $, naturally associated with the space of Bessel potentials of $L^p$-functions, then our principal result is the estimate: for $1< \alpha p\le n$, there is a constant $C = C(\alpha ,p,n)$ such that for any set $E$\begin{} \min \lbrace B_{\alpha ,p}(E\cap Q),B_{\alpha ,p}(E^ c\cap Q)\rbrace \le C\cdot B_{\alpha ,p}(Q\cap \partial _ fE) \end{}for all open cubes $Q$ in $n$-space. Here $\partial _f E$ is the boundary of the $E$ in the $(\alpha ,p)$-fine topology i.e. the smallest topology on $c$-space that makes the associated $(\alpha ,p)$-linear potentials continuous there. As a consequence, we deduce that for $\alpha p>1$, open connected sets are connected in the $(\alpha ,p)$-quasi topology (i.e. the topology generated by the set function $B_{\alpha ,p}$ in the sense of Fuglede), and the $(\alpha ,p)$-finely open $(\alpha ,p)$-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 $n=3$.

LA - eng

KW - nonlinear potential theory; fine topologies; quasi topologies; Bessel capacities; method of balayage

UR - http://eudml.org/doc/74668

ER -

## References

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