Page 547 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 547
TOPICS IN SUB-ELLIPTIC AND ELLIPTIC PDES (MS-31)
On the mean value formula for harmonic functions
Giovanni Cupini, giovanni.cupini@unibo.it
Università di Bologna, Italy
The mean integral of harmonic functions on a ball is equal to the value of the functions at the
center of the ball. This is the well known Gauss mean value formula for harmonic functions.
This formula is “stable” and provides a harmonic characterizion of balls. In the talk I will
discuss these and related results, obtained in collaboration with E. Lanconelli and with N. Fusco,
E. Lanconelli, X. Zhong.
The Dirichlet problem for fully nonlinear degenerate elliptic equations
with a singular nonlinearity
Giulio Galise, galise@mat.uniroma1.it
Sapienza Università di Roma, Italy
Coauthor: Isabeau Birindelli
We consider the homogeneous Dirichlet problem, in uniformly convex domains, for a large
class of degenerate elliptic equations with singular zero order term. In particular we establish
sharp existence and uniqueness results of positive viscosity solutions.
Time-dependent focusing Mean Field Games with strong aggregation
Daria Ghilli, daria.ghi88@gmail.com
University of Padua, Italy
Mean Field Games (MFG) theory models the behavior of an infinite number of indistinguish-
able rational agents aiming at minimising a common cost. A large part of MFG literature is
devoted to the study of MFG systems with increasing coupling ("non focusing"case). Heuristi-
cally, this assumption means that agents prefer sparsely populated areas (indeed concentration
costs), and it is well-suited to model competitive cases. Moreover, the increasing monotonicity
of the coupling ensures existence and regularity of solutions in many circumstances. We are
interested in the "focusing"case, that is, where the coupling is monotone decreasing and it is a
local function of the distribution, so that no regularising effect can be expected. These systems
describe Nash equilibria of games with a large number of agents aiming at aggregation. In this
talk, we will introduce the model in the focusing case and we will show that there is a threshold
for the growth of the coupling, after which the solutions to the MFG system may not exist. This
is coherent with the focusing character of the MFG, which induces solutions to concentrate and
develop singularities.
545
On the mean value formula for harmonic functions
Giovanni Cupini, giovanni.cupini@unibo.it
Università di Bologna, Italy
The mean integral of harmonic functions on a ball is equal to the value of the functions at the
center of the ball. This is the well known Gauss mean value formula for harmonic functions.
This formula is “stable” and provides a harmonic characterizion of balls. In the talk I will
discuss these and related results, obtained in collaboration with E. Lanconelli and with N. Fusco,
E. Lanconelli, X. Zhong.
The Dirichlet problem for fully nonlinear degenerate elliptic equations
with a singular nonlinearity
Giulio Galise, galise@mat.uniroma1.it
Sapienza Università di Roma, Italy
Coauthor: Isabeau Birindelli
We consider the homogeneous Dirichlet problem, in uniformly convex domains, for a large
class of degenerate elliptic equations with singular zero order term. In particular we establish
sharp existence and uniqueness results of positive viscosity solutions.
Time-dependent focusing Mean Field Games with strong aggregation
Daria Ghilli, daria.ghi88@gmail.com
University of Padua, Italy
Mean Field Games (MFG) theory models the behavior of an infinite number of indistinguish-
able rational agents aiming at minimising a common cost. A large part of MFG literature is
devoted to the study of MFG systems with increasing coupling ("non focusing"case). Heuristi-
cally, this assumption means that agents prefer sparsely populated areas (indeed concentration
costs), and it is well-suited to model competitive cases. Moreover, the increasing monotonicity
of the coupling ensures existence and regularity of solutions in many circumstances. We are
interested in the "focusing"case, that is, where the coupling is monotone decreasing and it is a
local function of the distribution, so that no regularising effect can be expected. These systems
describe Nash equilibria of games with a large number of agents aiming at aggregation. In this
talk, we will introduce the model in the focusing case and we will show that there is a threshold
for the growth of the coupling, after which the solutions to the MFG system may not exist. This
is coherent with the focusing character of the MFG, which induces solutions to concentrate and
develop singularities.
545
