# School Schedule

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## Lecture Abstracts:

Henrik Shahgholian: Semilinear problems with FB

Abstract: I shall discuss recent developments of free boundary regularity that arise in semi-linear problems, due to phase change caused by the non-linearity.

The lectures are aimed at a larger audience but some very basic PDE will be required to be able to follow the course.

John Andersson: Linearization techniques in PDE and free boundary problems

Supplementary Lecture Notes on The Obstacle Problem

Abstract: There has been a strong trend to analyse free boundaries by means of “geometric methods” such as controlling the level set of a solution by means of the gradient. These methods place strong structural demands on the equations analysed and can not be extended in an obvious way to general equations, free boundary problems or systems.

In this talk we will investigate an analytical approach to the regularity theory of free boundaries. The analytical method is more robust and applies to more problems as well as systems of equations.

In order to be able to keep the level of the talk at a (humiliatingly?) low level we will mostly (maybe only) talk about the simplest free boundary problem:

$$\begin{array}{ll} \Delta u(x)=\chi_{\{u>0\}} & \textrm{ in }B_1(0) \\ u(x)=f(x)\ge 0 & \textrm{ on } \partial B_1(0), \end{array}$$

where $\chi_{\{u>0\}}$ is the characteristic function (that is equal to 1 on the set where $u>0$ and zero else).

The talks will only assume a general undergraduate analysis background and directed towards a general audience.

Rafayel Barkhudaryan: Viscosity Solutions and Finite Difference Methods for Obstacle Type Problems

Abstract: TBA

Rafayel Teymurazyan: An overview of the obstacle problem

Absrtact: The obstacle problem is a classic example of a free boundary problem. The problem is to find the equilibrium position of an elastic membrane, which has a fixed boundary and lies above a given obstacle. It has applications in optimal control, financial mathematics etc.

We will discuss the problem in its simplest form. Regularity results (of the solution and the free boundary) will be presented.
The talks are aimed at a large audience and assume general undergraduate analysis background.

Avetik Arakelyan: Some Multi-phase free boundary problems (FBP) and their applications

Abstract: In a series of talks I shall discuss some free boundary problems, which have growing interest due to their important applications in applied mathematics. The first application is the FBP arising in the so-called Quadrature domains (QD), which refers to a generalized type of mean value property for harmonic functions. I will start with the well-established and developed theory of QD and show its generalization to a multi-phase counterpart. In particular, I will discuss possibilities of multi-junction points (see Figure).

The second application will be the numerical treatment of a general class of some obstacle-like FBP that arise in population dynamics. To be more precise, I will discuss the numerical approximation of equations
of stationary states for a general class of the spatial segregation of Reaction-diffusion systems with m>=2 population densities having disjoint supports.

Spring School on Nonlinear PDEs and FBPs, 2016 © 2016 Frontier Theme