Symmetrization results in electrostatic problems
Two electrostatic problems are analysed from the point of view of symmetrization. At first the p-capacity of a system of conductors is studied. Let \(E\subset \subset G\) be two bounded open subsets of \({\mathbb{R}}^ n\) and let \(H_ i\), \(i=1,...,m\), be m bounded open subsets of \({\mathbb{R}}^ n\) such that \(H_ i\cap H_ j=\emptyset\), \(\forall i\neq j\), and \(\cup_{i}H_ i\subset \subset (G\setminus E)\). The p-capacity is defined as \[ C_ p=\inf \{\int_{D\setminus E}a(x)| Dv|^ p dx:\;v=0\text{ on } \partial G,\quad v=1\text{ on } \partial E,\quad v=const.\text{ on } \partial H_ i,\quad \forall i\}, \] where \(D=G\setminus (\cup_{i}H_ i)\), a(x)\(\geq 0\), \(a^{-1}\in L^ r(D\setminus E)\), \(r>1\). It is proved that \(C_ p\) decreases under symmetrization, that is, \[ C_ p\geq C^*_ p\equiv \inf \{\int_{D^*\setminus E^*}a_*(x)| Dv|^ p dx:\;v=0\text{ on } \partial D^*,\quad v=1\text{ on } \partial E^*\}, \] where \(D^*\), \(E^*\) are the n-dimensional balls in \({\mathbb{R}}^ n\) centered at the origin and with the same measure as D and E, respectively, and \(a_*(x)\) is the spherically symmetric increasing rearrangement of a(x), defined in \(D^*\setminus E^*\). The proof is based, among other things, on isoperimetric estimates for the \(L^{\infty}\)-norm of functions whose gradient has fixed rearrangement. A similar result is also obtained for the electrostatic energy of the field generated by three conductors at fixed potential.
scientific article; zbMATH DE number 4169036