Fine and quasi connectedness in nonlinear potential theory

David R. Adams; John L. Lewis

Annales de l'institut Fourier (1985)

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

Abstract

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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 < α p ≤ n , there is a constant C = C ( α , p , n ) such that for any set E min { B α , p ( E ∩ Q ) , B α , p ( E c ∩ Q ) } ≤ C · B α , p ( Q ∩ ∂ f E ) 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 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 .

How to cite

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Adams, 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, $\alpha &gt;0$, $1&lt; p&lt; \infty $, naturally associated with the space of Bessel potentials of $L^p$-functions, then our principal result is the estimate: for $1&lt; \alpha p\le n$, 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 $\alpha p&gt;1$, 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, $\alpha &gt;0$, $1&lt; p&lt; \infty $, naturally associated with the space of Bessel potentials of $L^p$-functions, then our principal result is the estimate: for $1&lt; \alpha p\le n$, 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 $\alpha p&gt;1$, 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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