Measure-Valued solutions to a class of equations with variable parabolicity direction

Luogo:  Aula Seminari “-1” – Dipartimento di Matematica
Ore 15:00

• Relatore: Emanuele Bottazzi (Università di Trento - Dottorando Matematica)

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Abstract: 
We review some notions of measure-valued solutions for a class of equations with variable parabolicity direction, depending upon the choice of a nonlinear, nonmonotone function ϕ. These equations are-ill-posed forward in time; nevertheless, a notion of measure-valued solution has been proposed by Plotnikov [1] under the assumption that ϕ is "cubic-like", and by Smarrazzo [2] under the assumption that ϕ is of a "Perona-Malik type". After an introduction on the ill-posed problem and a discussion of the hypotheses over ϕ, we review the definition of the measure-valued solutions, we will outline the proof of their existence, and we will discuss some of their properties. In a subsequent seminar, that will take place on the 19th of December, we will show another notion of solution for this class of equations. This notion of solution is formulated in the setting of nonstandard analysis, and generalizes simultaneously the measure-valued solutions introduced by Plotnikov and by Smarrazzo.

[1] P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Differential Equations 30 (1994), pp. 614–622.
[2] F. Smarrazzo, On a class of equations with variable parabolicity direction, Discrete and continuous dynamical systems 22 (2008), pp. 729–758.

Referente: Stefano Baratella