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Zbl 0657.49005
Dal Maso, Gianni
$\Gamma$-convergence and $\mu$-capacities. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No.3, 423-464 (1987). ISSN 0391-173X

Given a bounded open subset $\Omega$ of ${\bbfR}\sp n$ (n$\ge 2)$, denote by ${\cal M}\sb 0(\Omega)$ the class of all non-negative Borel measures on $\Omega$ which vanish on all sets of (harmonic) capacity zero, but may take the value $+\infty$ on non-polar subsets of $\Omega$. For every $\mu\in {\cal M}\sb 0(\Omega)$ it is possible to define the associated $\mu$-capacity by setting $$ (1)\quad cap\sb{\mu}(B)=\inf\sb{u\in H\sp 1\sb 0(\Omega)}\{\int\sb{\Omega}\vert Du\vert\sp 2dx+\int\sb{B}(u-1)\sp 2d\mu \}. $$ In the paper the author studies the relations between the measures $\mu\in {\cal M}\sb 0(\Omega)$ and the corresponding capacities $cap\sb{\mu}$. A subclass ${\cal M}\sp*\sb 0(\Omega)\subset {\cal M}\sb 0(\Omega)$ is introduced, and for every $\mu\in {\cal M}\sp*\sb 0(\Omega)$ the capacity $cap\sb{\mu}$ defined in (1) is shown to be a Choquet capacity. Moreover, for every $\mu\in {\cal M}\sb 0(\Omega)$ there exists $\mu\sp*\in {\cal M}\sp*\sb 0(\Omega)$ which is equivalent to $\mu$ in the sense that $$ \int\sb{\Omega}u\sp 2d\mu\sp*=\int\sb{\Omega}u\sp 2d\mu \quad for\quad every\quad u\in H\sp 1\sb 0(\Omega). $$ A convergence (called $\gamma$-convergence) is introduced on ${\cal M}\sb 0(\Omega)$ in such a way that the solutions $u\sb h$ of the elliptic problems formally written as $$-\Delta u\sb h+\mu\sb hu\sb h=f\quad in\quad \Omega$$ converge in $L\sp 2(\Omega)$ (as $\mu\sb h$ $\gamma$-converges to $\mu)$ to the solution u of $$-\Delta u+\mu u=f\quad in\quad \Omega. $$
[G.Buttazzo]

MSC 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence
31B15 Potentials, etc. (higher-dimensional)
54A20 Convergence in general topology

Keywords: relaxed problems; Choquet capacity; $\gamma$-convergence

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