# Optimization problems governed by Allen-Cahn and Cahn-Hilliard variational inequalities

### From DFG-SPP 1253

## Project leader

NWF I - Mathematik

Uni Regensburg

93040 Regensburg

phone: 0941 943-2794

fax: 0941 943-3263

homepage: http://www.mathematik.uni-regensburg.de/Mat8/Blank/

Lehrstuhl für Mathematik VIII, NWF I – Mathematik

Uni Regensburg

Tel: 0941 943-2992

Fax: 0941 943-3263

Email: harald.garcke@mathematik.uni-regensburg.de

Homepage:

## DFG funded assistants

Hassan Farshbaf-Shaker (Uni Regensburg)

hassan.farshbaf-shaker@mathematik.uni-regensburg.de

## Description

The Allen-Cahn or Cahn-Hilliard variational inequalities can be derived as gradient flow, based on the Ginzburg-Landau energy with an obstacle potential. This project will be concerned with the multicomponent models resulting in vector valued problems. Starting from the gradient flow structure we will address the following main issues.

- Topology optimisation with non-local Allen-Cahn variational inequalities. In structural topology optimisation problem one tries to find the material distribution in a given design domain, such that the design objective is optimised. This project is mainly concerned with two aspects thereof, more precisely with:
- Mean compliance optimisation using Allen-Cahn variational inequalities, where the Ginzburg-Landau energy has to be extended by an elastic contribution.
- Relating phase field approaches to sharp interface approaches by means of asymptotic expansion techniques for phase field equations.

- Optimal control of Allen-Cahn and Cahn-Hilliard variational inequality systems using a force term in the equations. This results into MPEC problems in function spaces. Applications are for example the control of quantum dot formation or the morphing problem, where the goal is the transformation of a given interface into another one. We want to derive
- First order optimality conditions for the continuous setting.
- Efficient computational tools to solve the MPECs on the discrete level.
- A relation to an optimisation problem with the sharp interface model as constraint by means of asymptotic expansion techniques.