Monotonicity formulas in potential theory

Abstract

Using the electrostatic potential u due to a uniformly charged body Ω ⊂ R n , n ≥ 3, we introduce a family of monotone quantities associated with the level set flow of u. The derived monotonicity formulas are exploited to deduce a new quantitative version of the classical Willmore inequality.MSC (2010): 35B06, 53C21, 35N25.

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

“…convex) set is considered, see the appendix in [2]. This latter work generalizes previous Euclidean results [5] and is robust enough to be extended to the non-linear frame, see [3,14,32]. Beyond their intrinsic interest, the above monotonicity formulas have important consequences, for instance, in the study of tangent cones at infinities of Einstein manifolds [24], of the Minkowski formulas in convex geometry [2,3], and of the asymptotically flat static metrics in general relativity [4,15].…”

On the Mass of Static Metrics with Positive Cosmological Constant: II

“…The case of equality requires a different analysis, and will be studied separately in Section 8. The cylindrical ansatz is inspired by the analogous technique used in [2,3,4], and consists in an appropriate conformal change of the original triple. The idea comes from the observation that the Schwarzschild-de Sitter metric can be made cylindrical via a division by |x| 2 .…”

Geometric aspects of p-capacitary potentials

“…In other words, we are going to exploit both the local and the global features of our theorems. Let us also point out that the monotonicity of V p q allows us to extend to the nonlinear case all the results provided in [5], while the monotonicity of V p ∞ extends the results contained in [9]. In particular, we provide a unified approach to the type of estimates considered in these two papers.…”

“…Consequences of the main Theorems. In this section we mainly follow the scheme proposed in [5] to get various consequences of Theorem 1.1 and Theorem 1.3. More precisely, in the first subsection we use (1.4) and (1.6), to deduce various sharp inequalities involving u and Ω, while in the second subsection we will compare the value of our monotone functions on different level sets of u.…”

“…In particular, we provide a unified approach to the type of estimates considered in these two papers. Indeed, both the monotonicity of V p q and and V p ∞ will be proved in the conformal setting provided by the cylindrical ansatz, introduced in [5], while in [9] a different geometric approach named by the authors spherical ansatz was considered. Before proceeding notice that ∂Ω = {u = 1}, and consequently, if ν is the (interior) normal to ∂Ω, we get ∂u/∂ν = −|Du|.…”