Page 526 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 526
NONSMOOTH VARIATIONAL METHODS FOR PDES AND APPLICATIONS IN
MECHANICS (MS-8)
On hysteresis reaction-diffusion systems and application in population
dynamics
Klemens Fellner, klemens.fellner@uni-graz.at
University of Graz, Austria
We consider a general class of PDE-ODE reaction-diffusion systems, which exhibits a singular
fast-reaction limit towards a reaction-diffusion equation coupled to a scalar hysteresis operator.
As applicational motivation, we present a PDE model for the growth of a population accord-
ing to a given food supply coupled to an ODE for the turnover of a food stock. Under realistic
conditions the stock turnover is much faster than the population growth yielding an intrinsic
scaling parameter. We emphasise that the structural assumptions on the considered PDE-ODE
models are quite general and that analogue systems might describe e.g. cell-biological buffer
mechanisms, where proteins are stored and used at the same time.
Finally, we present a new kind of hysteresis-diffusion driven instability caused by the non-
linear coupling between a reaction-diffusion equation and a scalar generalised play operator.
We discuss in detail how this coupling with a generalised play operator can lead to spatially
inhomogeneous large-time behaviour or equilibration to a homogeneous state.
Lagrange multipliers and nonconstant gradient constrained problem
Sofia Giuffrè, sofia.giuffre@unirc.it
Mediterranea University of Reggio Calabria, Italy
The talk is aimed at studying a gradient constrained problem associated to a linear operator. This
classical problem was subject to an intense study a few decades ago (see [1, 2, 4, 6, 8], but some
very important issues were left open. In particular, we are able to prove two kinds of results (see
[5]): first, we prove the equivalence of a non-constant gradient constraint problem to a suitable
obstacle problem, where the obstacles solve a Hamilton-Jacobi equation in the viscosity sense
(see [7]) and, second, we obtain the existence of Lagrange multipliers associated to the problem.
The Lagrange multipliers exist as a Radon measure in the case that the free term of the equation
f ∈ Lp, p > 1, whereas, if f is a positive constant, it is possible to regularize the result, namely
to prove that they belong to L2. These results have been obtained, using a new theory of infinite
dimensional duality contained in [3]. The classical strong duality theory does not work in an
infinite dimensional setting, when the interior of the ordering cone of the sign constraints is
empty and this new theory overcomes this difficulty.
References
[1] H. Brezis, G. Stampacchia, Sur la régularité de la solution d’inéquations elliptiques, Bull.
Soc. Math. France 96 (1968) 153–180.
[2] G. Cimatti, The plane stress problem of Ghizetti in elastoplasticity, Appl. Math. Optim.3
(1), 15-26 (1976).
[3] P. Daniele, S. Giuffrè, A. Maugeri, F.Raciti, Duality theory and applications to unilateral
problems, J. Optim. Theory Appl. (2014) 162:718-734.
[4] L. Evans, A second order elliptic equation with gradient constraint, Comm.Part. Diff. Eq.,
4 (1979), 555-572.
524
MECHANICS (MS-8)
On hysteresis reaction-diffusion systems and application in population
dynamics
Klemens Fellner, klemens.fellner@uni-graz.at
University of Graz, Austria
We consider a general class of PDE-ODE reaction-diffusion systems, which exhibits a singular
fast-reaction limit towards a reaction-diffusion equation coupled to a scalar hysteresis operator.
As applicational motivation, we present a PDE model for the growth of a population accord-
ing to a given food supply coupled to an ODE for the turnover of a food stock. Under realistic
conditions the stock turnover is much faster than the population growth yielding an intrinsic
scaling parameter. We emphasise that the structural assumptions on the considered PDE-ODE
models are quite general and that analogue systems might describe e.g. cell-biological buffer
mechanisms, where proteins are stored and used at the same time.
Finally, we present a new kind of hysteresis-diffusion driven instability caused by the non-
linear coupling between a reaction-diffusion equation and a scalar generalised play operator.
We discuss in detail how this coupling with a generalised play operator can lead to spatially
inhomogeneous large-time behaviour or equilibration to a homogeneous state.
Lagrange multipliers and nonconstant gradient constrained problem
Sofia Giuffrè, sofia.giuffre@unirc.it
Mediterranea University of Reggio Calabria, Italy
The talk is aimed at studying a gradient constrained problem associated to a linear operator. This
classical problem was subject to an intense study a few decades ago (see [1, 2, 4, 6, 8], but some
very important issues were left open. In particular, we are able to prove two kinds of results (see
[5]): first, we prove the equivalence of a non-constant gradient constraint problem to a suitable
obstacle problem, where the obstacles solve a Hamilton-Jacobi equation in the viscosity sense
(see [7]) and, second, we obtain the existence of Lagrange multipliers associated to the problem.
The Lagrange multipliers exist as a Radon measure in the case that the free term of the equation
f ∈ Lp, p > 1, whereas, if f is a positive constant, it is possible to regularize the result, namely
to prove that they belong to L2. These results have been obtained, using a new theory of infinite
dimensional duality contained in [3]. The classical strong duality theory does not work in an
infinite dimensional setting, when the interior of the ordering cone of the sign constraints is
empty and this new theory overcomes this difficulty.
References
[1] H. Brezis, G. Stampacchia, Sur la régularité de la solution d’inéquations elliptiques, Bull.
Soc. Math. France 96 (1968) 153–180.
[2] G. Cimatti, The plane stress problem of Ghizetti in elastoplasticity, Appl. Math. Optim.3
(1), 15-26 (1976).
[3] P. Daniele, S. Giuffrè, A. Maugeri, F.Raciti, Duality theory and applications to unilateral
problems, J. Optim. Theory Appl. (2014) 162:718-734.
[4] L. Evans, A second order elliptic equation with gradient constraint, Comm.Part. Diff. Eq.,
4 (1979), 555-572.
524
