Some symmetry results for integral equations involving Wolff potential on bounded domains

Sharp pointwise gradient estimates for Riesz potentials with a bounded density

“…Remark also that the obtained gradient estimates are sharp for Euclidean balls with constant density. The latter symmetry phenomenon is natural for the Riesz and Newton potentials [6], [18], and studied recently in connection with Riesz potential integral equations on exterior domains [22], [14], [13], [24].…”

“…Using (51), Theorem 2.3 implies the following new inequality, see (13) and the discussion afterwards. Coming back to the Markov type inequality (14) involving s −1 , we obtain a further extension: in the notation of (9) we have (73) sinh 2 s −1 ≤ s 0 s −2…”

“…Remark that in contrast to the algebraic character of the classical Markov inequalities [5], (14) has a different, transcentental nature. The analogous two-dimensional L-problem is much less explored, recent works point out some direct applications of this problem to tomography, geophysics, the problem in particular has to do with the distribution of pairs of random variables or the logarithmic potential of a planar domain, see [11], [10], [16], [12].…”

Existence and nonexistence of positive solutions to an integral system involving Wolff potential

“…Several authors studied the special cases about (3) and (5). For example, Huang, Hong, Li [9] investigated system of integral equations involving Wolff potential on a bounded domain such as…”