36 research outputs found
Balayage and fractional harmonic measure in minimum Riesz energy problems with external fields
For the Riesz kernel on ,
where , , and , we consider the problem
of minimizing the Gauss functional
being a
given positive (Radon) measure on , and ranging over all
positive measures of finite energy, concentrated on and
having unit total mass. We prove that if is a quasiclosed set of nonzero
inner capacity , and if the inner balayage of onto
is of finite energy, then the solution to the problem
exists if and only if either , or . We also analyze the support of
, thereby discovering new surprising phenomena. To be precise,
we say that is inner -ultrairregular for
if , being the inverse of with respect to
; let consist of all those . We show that for any , there is
such that do exist for all
. Here , being the
unit Dirac measure at . Thus, for any and , no
compensation effect occurs between two oppositely signed charges carried by the
same conductor, which seems to contradict our physical intuition. Another
interesting phenomenon is that, if is closed while unbounded, then for any ,
is noncompact, whereas is
already compact for any -- even arbitrarily small.Comment: 24 page
Minimum Riesz energy problems with external fields
The paper deals with minimum energy problems in the presence of external
fields with respect to the Riesz kernels , , on
, . For quite a general (not necessarily lower
semicontinuous) external field , we obtain sufficient and/or necessary
conditions for the existence of minimizing the Gauss functional
over all positive
Radon measures with , concentrated on quite a general
(not necessarily closed) . We provide various alternative
characterizations of the minimizer , and analyze the continuity
of both and the modified Robin constant for monotone families
of sets. Finally, we establish a complete description of the support of
, thereby giving an answer to a question often discussed in the
literature. The results obtained present a significant extension of the theory
in question; crucial to this is a new approach based on the close interaction
between the strong and the vague topologies as well as on the theory of inner
balayage, developed recently by the author.Comment: 28 pages, 2 figures. arXiv admin note: text overlap with
arXiv:2207.1434