 Department of Mathematics and Systems Analysis

# Analysis and PDE seminars at Aalto

## Seminar on analysis and geometry

Analysis and geometry seminar is held usually on Wednesdays at 12-14 in M3

Organizers:

## Talks

• 29.1. 12:15  Marti Prats: Minimizers for the thin one-phase free boundary problem – M3 (M234)

We will give an overview of the literature on the non-negative minimizers for the one-phase free boundary problem of Alt and Caffarelli. This functional contains two competing terms, the standard Dirichlet energy and the measure of the set where the function is positive. Every minimizer is harmonic in its positive phase and vanishes elsewhere. Many questions arise regarding the regularity of the free boundary of such a minimizer, some of them still open. We will also discuss how these ideas can be brought to the thin one-phase free boundary problem, where the first term is a weighted Dirichlet energy related to the Poisson extension used to compute the fractional Laplacian, and the second competing term is only evaluated in a hyperplane. Minimizers of such a functional will have vanishing fractional Laplacian in the hyperplane's positive phase. This intrinsic nonlocallity will make some arguments to vary substantially.

• 5.2. 12:15  Prashanta Garain: TBA – M3 (M234)
• 12.2. 12:15  Matias Vestberg: TBA – M3 (M234)

We will give an overview of the literature on the non-negative minimizers for the one-phase free boundary problem of Alt and Caffarelli. This functional contains two competing terms, the standard Dirichlet energy and the measure of the set where the function is positive. Every minimizer is harmonic in its positive phase and vanishes elsewhere. Many questions arise regarding the regularity of the free boundary of such a minimizer, some of them still open. We will also discuss how these ideas can be brought to the thin one-phase free boundary problem, where the first term is a weighted Dirichlet energy related to the Poisson extension used to compute the fractional Laplacian, and the second competing term is only evaluated in a hyperplane. Minimizers of such a functional will have vanishing fractional Laplacian in the hyperplane's positive phase. This intrinsic nonlocallity will make some arguments to vary substantially.

• 26.2. 12:15  Peter Lindqvist (NTNU): TBA – M3 (M234)

We will give an overview of the literature on the non-negative minimizers for the one-phase free boundary problem of Alt and Caffarelli. This functional contains two competing terms, the standard Dirichlet energy and the measure of the set where the function is positive. Every minimizer is harmonic in its positive phase and vanishes elsewhere. Many questions arise regarding the regularity of the free boundary of such a minimizer, some of them still open. We will also discuss how these ideas can be brought to the thin one-phase free boundary problem, where the first term is a weighted Dirichlet energy related to the Poisson extension used to compute the fractional Laplacian, and the second competing term is only evaluated in a hyperplane. Minimizers of such a functional will have vanishing fractional Laplacian in the hyperplane's positive phase. This intrinsic nonlocallity will make some arguments to vary substantially.

• 18.3. 12:15  Sari Rogovin: Poincaré inequalities and a general quasihyperbolic growth condition – M3 (M234)
• 25.3. 12:15  Emanuel Carneiro (ICTP Trieste): TBA – M3 (M234)

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