243298 Uniqueness results for elliptic and parabolic problems (S) (SoSe 2025)

This seminar will focus on uniqueness results to degenerate
elliptic and parabolic partial differential equations.
We will begin with uniqueness in suitable spaces for solutions to
linear equations both on bounded and unbounded domains.
- A unified theory for boundary value problems of second order
elliptic equations, see Fichera, Editore: Univ. Of Wisconsin
Press, Anno edizione: 1960
- Uniqueness in suitable weighted L^1 and L^2 spaces for weak
solutions to Cauchy problems with linear diffusion, see
Aronson-Besala, J. Math. Anal. Appl. 13(3), 1966, 516-526.

Afterwards, we will address similar questions in the framework of
nonlinear diffusions, such as the case of the porous medium
equation.
- Cauchy problems for the porous medium equations on bounded
domains and bounded data, see _D.G. Aronson, M.G. Crandal, L.A.
Peletier_, J. Nonlinear Anal. 6, 1982, 1001-1022
- Cauchy problems with non-linear diffusion on unbounded domains
and bounded data, see Kamin, Kersner, Tesei when n=1 Rendiconti
Lincei, 1998; Punzo, J. Evol. Eq. 2009 when n>=2
- Cauchy problems for the porous medium equation on unbounded
domains and unbounded data, see P. Bénilan, M.G. Crandall, M.
Pierre, Indiana Univ. Math. J. 33 (1984) 51–87.

We will adjust the specific topics in order to match the
participant's levels and interests.

Prerequisites:
Functional analysis and a course in PDE

Preliminary meeting:
31.01.2025 , 11:00 in V5-148