Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization

Abstract

The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.Dedicated to Guido Weiss, with gratitude for his encouragement, and appreciation of his far-reaching vision in Analysis.

Two balls maximize the third Neumann eigenvalue in hyperbolic space

“…by ( 23), ( 24) and (25) . Integrating over Ω gives zero (as desired) on the right side, thanks to condition (22) for the center of mass c p,0 .…”

“…and that this unique point c H depends continuously on the parameters (p, t) of the halfspace H. These claims follow from [22, Corollary 3] (with f ≡ 1 there); in the conclusion of that result, take c H = −x H , and notice the hypotheses of the corollary are satisfied because g is increasing with g(r) > 0 when r > 0. Incidentally, results in [22] can also handle unbounded sets of finite volume, if desired. We call c H the "center of mass point" corresponding to the halfspace H and, as in the case of the fold map defined above, will use the notation c H or c p,t depending on the context.…”

“…To prove c −p,0 = R p (c p,0 ), we must show that R p (c p,0 ) satisfies the condition determining the unique point c −p,0 , namely condition (22) with p and t replaced by −p and 0. That is, we must show…”

“…Hence ˆΩ v τ 0 • Φ p,τ (y) dγ(y) = ˆΩ v(y) dγ(y) = 0 by the normalization (21). Therefore the center of mass relation (22) holds with c p,τ = 0. Hence…”

“…Estimating the Rayleigh quotient. Fix the parameters (p, t) found in Proposition 7.3, and write H = H p,t for the hyperbolic halfspace, so that by the proposition and the earlier formula (22), the vector field v H (y) = v T −c H • Φ H (y) is orthogonal in L 2 (Ω; dγ) to both the constant function and the first excited state f . Substituting each component v H,1 , .…”