| Abstract: |
| Let $D$ be a bounded open set in $\mathbb{R}^n$. We consider the minimization and the maximization of the functional $ \Phi_s(f)= \frac{1}{2} \int \int_{\mathbb{R}^{2n}} \frac{(u_f(x)-u_f(y))^2}{|x-y|^{n+2s}}dxdy, $ where $u_f$ is the unique solution of the equation $(-\Delta)^s u =f$ in $D$, $u=0$ in $D^c$, over the convex closed set $ \left\{f\in L^\infty(D)\colon 0\leq f \leq 1,\,\,\int_D fdx=\beta \right\}. $ \bigskip We show the existence of the unique minimizer $f_{min}$ and a maximizer $f_{max}$. Moreover, for some constants $\alpha_{min}$ and $\alpha_{max}$ the functions $u_{min}=\alpha_{min}-u_{f_{min}}$ and $u_{max}=u_{f_{max}}$ solve the following equations in $D$ $$ -(-\Delta)^s u_{min}-\chi_{\{u_{min}\leq 0\}}\min\{-(-\Delta)^s u_{min}^+;1\}=\chi_{\{u_{min}>0\}}, $$ $$(-\Delta)^s u_{max}=\chi_{\{u_{max}>\alpha_{max}\}}.$$ |
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