Sumários

Minimisation of Convex Functions and Variational Inequalities

11 Abril 2019, 14:30 • José Francisco da Silva Costa Rodrigues

Minimisation of Convex Functions in reflexive Banach spaces—the direct method in the bonded case and in the coercive case. Characterisation of the solutions under (Gâteaux) differentiability conditions and in the general subdifferential case. Variational inequalities and the Minty trick. Proximity mappings in Hilbert spaces. The case of a bilinear symmetric form. The finite dimension approximation method of the existence of solutions of certain variational inequalities for monotone, coercive operators with different types of weak continuities: semi-continuous, demicontinuos and weakly continuous over the subspaces of finite dimension. Extension to pseudo-monotone operators. Chap. 2 of [ET].Further readings:
Brezis, Haïm Équations et inéquations non linéaires dans les espaces vectoriels en dualité. (French)  Ann. Inst. Fourier (Grenoble)  18 1968 fasc. 1, 115–175. http://www.numdam.org/item/?id=AIF_1968__18_1_115_0
Chapter 2 of Tomáš Roubí􀃾ek, Nonlinear Partial Differential Equations with Applications, Birkhauser, 2013

Minimisation of Convex Functions and Variational Inequalities

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

Minimisation of Convex Functions in reflexive Banach spaces—the direct method in the bonded case and in the coercive case. Characterisation of the solutions under (Gâteaux) differentiability conditions and in the general subdifferential case. Variational inequalities and the Minty trick. Proximity mappings in Hilbert spaces. The case of a bilinear symmetric form. The finite dimension approximation method of the existence of solutions of certain variational inequalities for monotone, coercive operators with different types of weak continuities: semi-continuous, demicontinuos and weakly continuous over the subspaces of finite dimension. Extension to pseudo-monotone operators. Chap. 2 of [ET].Further readings:
Brezis, Haïm Équations et inéquations non linéaires dans les espaces vectoriels en dualité. (French)  Ann. Inst. Fourier (Grenoble)  18 1968 fasc. 1, 115–175. http://www.numdam.org/item/?id=AIF_1968__18_1_115_0
Chapter 2 of Tomáš Roubí􀃾ek, Nonlinear Partial Differential Equations with Applications, Birkhauser, 2013

Minimisation of Convex Functions and Variational Inequalities

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

Minimisation of Convex Functions in reflexive Banach spaces—the direct method in the bonded case and in the coercive case. Characterisation of the solutions under (Gâteaux) differentiability conditions and in the general subdifferential case. Variational inequalities and the Minty trick. Proximity mappings in Hilbert spaces. The case of a bilinear symmetric form. The finite dimension approximation method of the existence of solutions of certain variational inequalities for monotone, coercive operators with different types of weak continuities: semi-continuous, demicontinuos and weakly continuous over the subspaces of finite dimension. Extension to pseudo-monotone operators. Chap. 2 of [ET].Further readings:
Brezis, Haïm Équations et inéquations non linéaires dans les espaces vectoriels en dualité. (French)  Ann. Inst. Fourier (Grenoble)  18 1968 fasc. 1, 115–175. http://www.numdam.org/item/?id=AIF_1968__18_1_115_0
Chapter 2 of Tomáš Roubí􀃾ek, Nonlinear Partial Differential Equations with Applications, Birkhauser, 2013

Introduction to Convex Analysis in Banach Spaces

4 Abril 2019, 14:30 • José Francisco da Silva Costa Rodrigues

Introduction to Convex Analysis in Banach Spaces (first Chap of [ET])


Convex sets and their separation— revision the Hahn-Banach theorem: geometrical and analytical forms; their first consequences in lcs and in normed spaces. Convex functions: effective domain; epigraph; lower semi-continuity; sufficient conditions for continuity. The Gamma(V) set. Conjugate (or polar) functions and convex duality. Subdifferentiability and relation with Gâteaux differentiability. Elements of sub differential calculus.

PS: Lecture from 14:30 til 17:15.

[ET] I.Ekeland & R.Témam—Convex Analysis and Variational Problems, SIAM Classics #28, 1999.

No lecture scheduled

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

No lecture scheduled