Si dà avviso che la prima lezione del corso tenuto dal prof. Ricardo H. Nochetto
Geometric Partial Differential Equations: Theory and Approximation
nell’ambito del progetto ‘Collegiale non residente’ dell’Università di Pavia
si terrà martedì 23/5 alle ore 16:15 presso il Collegio Nuovo.
Nella prima lezione verrà presentata una introduzione al Corso, e verranno fissati in accordo con gli studenti gli incontri successivi.
Qui sotto potete trovare un programma del corso:
————————– descrizione breve del corso ———————————-
Geometric Partial Differential Equations: Theory and Approximation
The purpose of this course is to discuss elements of differential geometry
in the context of analysis and approximation of geometric partial differential
equations (PDEs). This includes the study of variations of functionals with
respect to shape and applications to several geometric flows, finite element
methods for the Laplace-Beltrami operator, nonlinear plate theory and liquid
crystals. The emphasis is on variational formulations, approximation, and
Gamma-convergence.
———————————– programma del corso ————————————
Geometric Partial Differential Equations: Theory and Approximation
1. Introduction
Shape differential calculus: examples
Geometric gradient flows
2. Elements of Differential Geometry
Parametric surfaces: parametrizations, normal, area element
Tangential differential operators
Signed distance function
First and second fundamental forms
Divergence theorem on surfaces
The Laplace-Beltrami operator
3. Shape Differential Calculus
The velocity method
Material and shape derivatives
Shape derivatives of domain and contour integrals
Shape derivatives of geometric quantities
Shape derivatives of solutions of boundary value problems
4. Finite Element Methods for the Laplace-Beltrami Operator
Parametric FEM
Trace FEM
Narrow band FEM
5. Geometric Gradient Flows
Motivation: Allen-Cahn and Cahn-Hilliard models
Mean curvature flow
Optimal shape design
Surface diffusion
Willmore flow
Biomembranes: Helfrich flow
6. Gamma-Convergence
Definition
Convergence of absolute minimizers
Example: model reduction
7. Nonlinear Plate Theory
Nonlinear Kirchhoff model: large deformations and isometries
Bilayer plates
FEMs for bilayers
Gamma-convergence
Discrete gradient flow
8. Director Fields and Liquid Crystals
Approximation of director fields
Ericksen model for liquid crystals
Landau – de Gennes model for liquid crystals
Gamma-convergence
Discrete gradient flow