Geometric aspects in linear and nonlinear potential theory

Starting date
March 11, 2019
Duration (months)
12
Departments
Computer Science
Managers or local contacts
Agostiniani Virginia

The aim of the project is to shed light on some relevant geometric aspects arising in potential theory. In concrete, we are interested in studying geometric properties of a domain in connection with the related capacitary or p-capacitary potential, both in the Euclidean and in a more general Riemannian context.
The archetypical case is the one analysed in [2], where some new monotonicity formulas associated with the level set flow of the electrostatic potential in the Euclidean space are deduced. Such formulas are then applied to deduce a new quantitative estimate of the Willmore Inequality.
This point of view and the developed technique have already been fruitfully employed to various different contexts: to that of static metrics in general relativity [3,4,5], to the case of manifolds with nonnegative Ricci curvature [1], to the case of the Euclidean p-capacitary potential [6].
Here are, in a nutshell, the problems which we would like to consider in our project: Minkowski inequality for mean convex domains; Riemannian Penrose Inequality in low dimension; Monotonicity formulas in Cartan-Hadamard manifolds; Torsion problem with boundary made of more than one connected components; An alternative approach to the Willmore Conjecture.

References:

[1] V. Agostiniani, M. Fogagnolo, and L. Mazzieri. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature. arXiv:1812.05022.

[2] V. Agostiniani and L.Mazzieri. Monotonicity formulas in potential theory. arXiv:1606.02489v3.

[3] V. Agostiniani and L. Mazzieri. On the geometry of the level sets of bounded static potentials. Commun. Math. Phys., 355: 261-301, 2017.

[4] S. Borghini and L. Mazzieri. On the mass of static metrics with positive cosmological constant: II. arXiv:1711.07024.

[5] S. Borghini and L. Mazzieri. On the mass of static metrics with positive cosmological constant: I. Classical Quantum Gravity, 35(12): 125001, 43, 2018.

[6] M. Fogagnolo, L. Mazzieri, and A. Pinamonti. Geometric aspects of p-capacitary potentials. To appear on Ann. Inst. H. Poincaré Anal. Non Linéaire. https://doi.org/10.1016/j.anihpc.2018.11.005.

Sponsors:

INdAM
Funds: assigned and managed by an external body

Project participants

Virginia Agostiniani
Temporary Assistant Professor

Collaboratori esterni

Lorenzo Mazzieri
Università degli Studi di Trento
Andrea Pinamonti
Università degli Studi di Trento
Mattia Fogagnolo
Università degli Studi di Trento
Research areas involved in the project
Matematica - applicazioni e modelli
Manifolds

Activities

Research facilities