## Differential and Integral Equations

- Differential Integral Equations
- Volume 20, Number 6 (2007), 657-670.

### Hölder continuity in a shape-optimization problem with perimeter

#### Abstract

Here, we prove Hölder continuity of the state function in a shape-optimization problem arising, for instance, in the electromagnetic shaping of a liquid metal. The equilibrium state is obtained as minimizing the total energy, which is given in the form $$ \mathcal{E}_\lambda(\Omega) = J(\Omega) + P(\Omega) + \lambda ||\Omega| - m| , $$ where $\Omega$ is the domain occupied by the liquid, $P(\Omega)$ and $|\Omega|$ being respectively the perimeter and the Lebesgue measure of $\Omega$. We prove that the state function associated with the optimal shape is $\frac{1}{2}$-H\"older continuous, the main difficulty coming from the fact that the state function is not assumed to be non-negative.

#### Article information

**Source**

Differential Integral Equations, Volume 20, Number 6 (2007), 657-670.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356039430

**Mathematical Reviews number (MathSciNet)**

MR2319460

**Zentralblatt MATH identifier**

1212.35512

**Subjects**

Primary: 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]

Secondary: 35B65: Smoothness and regularity of solutions 35R35: Free boundary problems 49N60: Regularity of solutions

#### Citation

Landais, Nicolas. Hölder continuity in a shape-optimization problem with perimeter. Differential Integral Equations 20 (2007), no. 6, 657--670. https://projecteuclid.org/euclid.die/1356039430