View abstract

Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Wednesday, July 21, 19:00 ~ 19:30 UTC-3

A viscosity solution approach to regularity properties of the optimal value function

Pablo Ochoa

CONICET-Universidad Nacional de Cuyo, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak1d3412f42c3285b2c615be111d9016fb').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy1d3412f42c3285b2c615be111d9016fb = 'p&#97;bl&#111;.&#111;ch&#111;&#97;' + '&#64;'; addy1d3412f42c3285b2c615be111d9016fb = addy1d3412f42c3285b2c615be111d9016fb + '&#105;ng&#101;n&#105;&#101;r&#105;&#97;' + '&#46;' + '&#117;nc&#117;y&#111;' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_text1d3412f42c3285b2c615be111d9016fb = 'p&#97;bl&#111;.&#111;ch&#111;&#97;' + '&#64;' + '&#105;ng&#101;n&#105;&#101;r&#105;&#97;' + '&#46;' + '&#117;nc&#117;y&#111;' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloak1d3412f42c3285b2c615be111d9016fb').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy1d3412f42c3285b2c615be111d9016fb + '\'>'+addy_text1d3412f42c3285b2c615be111d9016fb+'<\/a>';

In this paper, we analyze the optimal value function $v$ associated to a general parametric optimization problem via the theory of viscosity solutions. The novelty is that we obtain regularity properties of $v$ by showing that it is a viscosity solution to a set of first-order equations. As a consequence, in Banach spaces, we provide sufficient conditions for local and global Lipschitz properties of $v$. We also derive, in finite dimensions, conditions for optimality through a comparison principle. Finally, we study the relationship between viscosity and Clarke generalized solutions to get further differentiability properties of $v$ in Euclidean spaces.

Joint work with Virginia N. Vera de Serio (Universidad Nacional de Cuyo, Argentina).