Sumários

8 Maio 2019, 17:00 • José Francisco da Silva Costa Rodrigues

Lagrangian and Saddle Points —relations with the primal and with the dual problems in the case of stable and normal assumptions; the case of reflexive Banach spaces.

The important special cases of primal and dual problems in ordered spaces under inequality constraints: the general abstract case; the specific form of the Lagrangian; the Kuhn-Tucker conditions in R^N;  the case of variational inequalities.
The bridal problem and generalised solutions.

Applied topics to be developed:
— non-differential functionals of the gradient— Bingham flow (special steady case) and the least-plastic torsion problem;
— non-differential functional in a nonlocal case — a problem in filtering theory;
— a non reflexive case— non-parametric minimal surfaces.

8 Maio 2019, 15:30 • José Francisco da Silva Costa Rodrigues

Lagrangian and Saddle Points —relations with the primal and with the dual problems in the case of stable and normal assumptions; the case of reflexive Banach spaces.

The important special cases of primal and dual problems in ordered spaces under inequality constraints: the general abstract case; the specific form of the Lagrangian; the Kuhn-Tucker conditions in R^N;  the case of variational inequalities.
The bridal problem and generalised solutions.

Applied topics to be developed:
— non-differential functionals of the gradient— Bingham flow (special steady case) and the least-plastic torsion problem;
— non-differential functional in a nonlocal case — a problem in filtering theory;
— a non reflexive case— non-parametric minimal surfaces.

Duality in Convex Optimisation

2 Maio 2019, 14:30 • José Francisco da Silva Costa Rodrigues

Motivation: formulation of the Primal and the Dual problem in the example of de Dirichlet integral associated with the Dirichlet problem for the Poisson equation—example of the conjugate function; minimisation of the solution versus maximising the symmetric of its gradient.Formulation of the abstract Primal and the Dual problem in the duality framework of convex functionals and sufficient conditions for the close relations between the two problems, for the existence of solutions and the respective extremality equations in general reflexive Banach spaces. Concretisation in important special cases in the Calculus of Variations. Examples: the linear Neumann problem; the nonlinear Dirichlet problem of p-Laplacian type; the Stokes problem; and the linear problem in elastostatics.References: Chap. 3 and 4 of Ekeland-Teman and the Duvaut-Lions' book for the Dual formulation in linear elasticity in page 119 and its relation with "Lagrange multipliers" in page 120 (see the connections of page 67 of Ekeland-Teman book). — The challenge is to put this system of linear elasticity in the duality framework, i.e., a special case of the general abstract theory of duality in convex analysis. (suggestion: start with the full Dirichlet homogeneous boundary data).


This lecture took place the Friday, 3-5-2019, at 14:00-17:00.

Operators in the Calculus of Variations

24 Abril 2019, 17:00 • José Francisco da Silva Costa Rodrigues

General variational inequalities for operators in the Calculus of Variations—existence in the case of pseudomonotone and coercive operators. Discussion of the concept of pseudomonotonicity and its relations with monotonicity, semicontinuity and the M-property. Their role in the proof of the existence of a solution to general variational inequalities by approximation of solutions in finite dimensional subspaces—extension of the Galerkin method. Examples of potential operators in reflexive Banach spaces and operators of the type of the Calculus of variations in the sense of Leray-Lions and their applications to quasi-linear elliptic operators with certain symmetry and growth properties of their coefficients.


Further reading: [L69]-J.L.Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires, Junod, Paris, 1969 (namely Sections 2 and 8 of Chap. 2)

Exercises 2.83 (page 75), 4.22+4.23+4.24 (page127) of Roubicek's 2013 book.

Lecture: 26-4-2919—14:00/17:30

Operators in the Calculus of Variations

24 Abril 2019, 15:30 • José Francisco da Silva Costa Rodrigues

General variational inequalities for operators in the Calculus of Variations—existence in the case of pseudomonotone and coercive operators. Discussion of the concept of pseudomonotonicity and its relations with monotonicity, semicontinuity and the M-property. Their role in the proof of the existence of a solution to general variational inequalities by approximation of solutions in finite dimensional subspaces—extension of the Galerkin method. Examples of potential operators in reflexive Banach spaces and operators of the type of the Calculus of variations in the sense of Leray-Lions and their applications to quasi-linear elliptic operators with certain symmetry and growth properties of their coefficients.


Further reading: [L69]-J.L.Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires, Junod, Paris, 1969 (namely Sections 2 and 8 of Chap. 2)

Exercises 2.83 (page 75), 4.22+4.23+4.24 (page127) of Roubicek's 2013 book.

Lecture: 26-4-2919—14:00/17:30