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G. Buttazzo
Italy

 

SHAPE OPTIMIZATION PROBLEMS

VIA MONGE-KANTOROVICH EQUATION



Giuseppe Buttazzo
(buttazzo@dm.unipi.it)


Dipartimento di Matematica
Via Buonarroti, 2
56127 PISA (Italy)




We consider the optimization problem

\begin{displaymath}\max\big\{{\cal E}(\mu)\ :\ \mu\hbox{ nonnegative measure, } \int d\mu=m\big\}\,,\end{displaymath}

where ${\cal E}(\mu)$ is the energy associated to $\mu$:

\begin{displaymath}{\cal E}(\mu)=\inf\big\{{1\over2}\int\vert Du\vert^2\,d\mu-\langle f,u\rangle\ : \ u\in{\cal D}({\bf R}^n)\big\}.\end{displaymath}

The datum $f$ is a signed measure with finite total variation and zero average. We show that the optimization problem above admits a solution which is not in $L^1({\bf R}^n)$ in general. This solution comes out by solving a Monge-Kantorovich equation of the form

\begin{displaymath}\cases{-\mathop {\rm div}\nolimits \big(\mu D_\mu w\big)=f\cr... ...p}_1({\bf R}^n),\quad\vert D_\mu w\vert=1\hbox{ $\mu$-a.e.}\cr}\end{displaymath}

where ${\rm Lip}_1({\bf R}^n)$ is the class of all Lipschitz functions on ${\bf R}^n$ with constant 1.


 

 

 

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Last modified: December 20, 1999