On the structure of partial balayage

Partial balayage on Riemannian manifolds

“…The paper builds on, and is inspired by, many previous papers and books in the area, for example (with an incomplete list, and partly repeating from the beginning of the introduction) D. Zidarov [52], M. Sakai [42], A. Varchenko, P. Etingof [50], H.S. Shapiro [47], E. Saff, V. Totik [40], H. Hedenmalm, S. Shimorin, N. Makarov [29,31], T. Sjödin, S. Gardiner [19,20,48], L. Levine, Y. Peres [34], F. Balogh, J. Harnad [2].…”

Convexity and the Exterior Inverse Problem of Potential Theory

“…The nonnegative function U µ − U D vanishes on D c , by (8), and hence on R N , since it is subharmonic on D. Thus (9) µ = λ | D and D ⊆ Ω(µ).…”

“…If µ is a measure with compact support, it is easy to see that there is a greatest subharmonic minorant s µ , say, of U µ + q on R N (using Theorem 3.7.5 of [2], for example). We need the following facts (see [3], [8]). …”

Classical and Stochastic Laplacian Growth

“…For details on partial balayage we refer to [243,238,526,201,483]. For details on partial balayage we refer to [243,238,526,201,483].…”

“…Indeed, (3.28) is true in Ω by definition (3.21) of Ω, and outside Ω we have u = 0, hence Δu = 0 there, at least under some regularity assumptions (e.g., μ ∈ L ∞ ); the general case can be handled by an approximation argument [526,201]. Thus Bal (μ, ρ) = μ outside Ω.…”