Page 224 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 224
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
provide sufficient conditions for the existence of solutions, in a suitable weak sense, satisfying
general boundary conditions expressed by means of a functional term H.
Second order necessary conditions for PDEs optimal control problems
with state constraints
Elsa Maria Marchini, elsa.marchini@polimi.it
Politecnico di Milano, Italy
This talk is devoted to second order necessary optimality conditions for optimal control prob-
lems under pure state constraints together with end point constraints, involving PDEs. Using
tools of second order variational analysis, we derive necessary optimality conditions in the form
of a maximum principle and a second order variational inequality. We further propose sufficient
conditions guaranteeing normality of the maximum principle. Applications to heat and wave
models will be proposed.
The results have been obtained jointly with Hélène Frankowska and Marco Mazzola
A fixed point approach for decaying solutions of difference equations
Serena Matucci, serena.matucci@unifi.it
University of Florence, Italy
Coauthors: Zuzana Došlá, Mauro Marini
A boundary value problem associated to the difference equation with advanced argument
∆ anΦ(∆xn) + bnΦ(xn+p) = 0, n ≥ 1 (∗)
is presented, where Φ(u) = |u|αsgn u, α > 0, p is a positive integer and the sequences a, b, are
positive. We deal with a particular type of decaying solutions of (∗), the so-called intermediate
solutions, that is solutions x of (∗) such that xn > 0, ∆xn < 0 for large n and
lim xn = 0, lim xn[1] = anΦ(∆xn) = −∞,
n n
where x[1] is called the quasidifference of x. In particular, we prove the existence of these
type of solutions for (∗) by reducing it to a suitable boundary value problem associated to a
difference equation without deviating argument. Our approach is based on a fixed point result
for difference equations, which originates from existing ones stated in the continuous case, but
take into account some peculiarities of the discrete case.
A Neumann p-Laplacian problem on metric spaces
Antonella Nastasi, ella.nastasi.93@gmail.com
University of Palermo, Italy
We use a variational approach to study existence and regularity of solutions for a Neumann
p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure
and supporting a Poincare’ inequality. Trace theorems for functions with bounded variation are
222
provide sufficient conditions for the existence of solutions, in a suitable weak sense, satisfying
general boundary conditions expressed by means of a functional term H.
Second order necessary conditions for PDEs optimal control problems
with state constraints
Elsa Maria Marchini, elsa.marchini@polimi.it
Politecnico di Milano, Italy
This talk is devoted to second order necessary optimality conditions for optimal control prob-
lems under pure state constraints together with end point constraints, involving PDEs. Using
tools of second order variational analysis, we derive necessary optimality conditions in the form
of a maximum principle and a second order variational inequality. We further propose sufficient
conditions guaranteeing normality of the maximum principle. Applications to heat and wave
models will be proposed.
The results have been obtained jointly with Hélène Frankowska and Marco Mazzola
A fixed point approach for decaying solutions of difference equations
Serena Matucci, serena.matucci@unifi.it
University of Florence, Italy
Coauthors: Zuzana Došlá, Mauro Marini
A boundary value problem associated to the difference equation with advanced argument
∆ anΦ(∆xn) + bnΦ(xn+p) = 0, n ≥ 1 (∗)
is presented, where Φ(u) = |u|αsgn u, α > 0, p is a positive integer and the sequences a, b, are
positive. We deal with a particular type of decaying solutions of (∗), the so-called intermediate
solutions, that is solutions x of (∗) such that xn > 0, ∆xn < 0 for large n and
lim xn = 0, lim xn[1] = anΦ(∆xn) = −∞,
n n
where x[1] is called the quasidifference of x. In particular, we prove the existence of these
type of solutions for (∗) by reducing it to a suitable boundary value problem associated to a
difference equation without deviating argument. Our approach is based on a fixed point result
for difference equations, which originates from existing ones stated in the continuous case, but
take into account some peculiarities of the discrete case.
A Neumann p-Laplacian problem on metric spaces
Antonella Nastasi, ella.nastasi.93@gmail.com
University of Palermo, Italy
We use a variational approach to study existence and regularity of solutions for a Neumann
p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure
and supporting a Poincare’ inequality. Trace theorems for functions with bounded variation are
222
