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In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity of the Newtonian potential of a bounded convex domain in ℝn , n ≥ 3, namely: if the Newtonian potential of a bounded domain is ”sufficiently concave”, then the domain is necessarily a ball. This result can be considered an unconventional overdetermined problem.This paper is based on a talk given by the author in Bologna at the ”Bruno Pini Mathematical Analysis Seminar”, which in turn was based on the paper P. Salani, A characterization of balls through optimal concavity for potential functions, Proc. AMS 143 (1) (2015), 173-183.

Metric theories of gravity are studied, beginning with a general action that is quadratic in curvature and allows arbitrary inverse powers of the d' Alembertian operator, resulting in infrared non-local extensions of general relativity. The field equations are derived in full generality and their consistency is checked by verifying the Bianchi identities. The weak-field limit is computed and a straightforward algorithm is presented to infer the post-Newtonian corrections directly from the action. This is then applied to various infrared gravity models including non-local Rf (R/square) dark energy and non-local massive gravity models. Generically, the Newtonian potentials are not identical and deviate from the 1/r behaviour at large distances. However, the former does not occur in a specific class of theories that does not introduce additional degrees of freedom in flat spacetime. A new non-local model within this class is proposed, defined by the exponential of the inverse d' Alembertian. This model exhibits novel features, such as the weakening of the gravity in the infrared, suggesting de-gravitation of the cosmological constant., QC 20150210

Metric theories of gravity are studied, beginning with a general action that is quadratic in curvature and allows arbitrary inverse powers of the d' Alembertian operator, resulting in infrared non-local extensions of general relativity. The field equations are derived in full generality and their consistency is checked by verifying the Bianchi identities. The weak-field limit is computed and a straightforward algorithm is presented to infer the post-Newtonian corrections directly from the action. This is then applied to various infrared gravity models including non-local Rf (R/square) dark energy and non-local massive gravity models. Generically, the Newtonian potentials are not identical and deviate from the 1/r behaviour at large distances. However, the former does not occur in a specific class of theories that does not introduce additional degrees of freedom in flat spacetime. A new non-local model within this class is proposed, defined by the exponential of the inverse d' Alembertian. This model exhibits novel features, such as the weakening of the gravity in the infrared, suggesting de-gravitation of the cosmological constant., AuthorCount:4