| Abstract: |
| We consider a functional with with the nonlocal obstacle acting on the function $V(x')=\int_0^1 U(x', t) dt $
$$
\int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x')^+\,dx'.
$$
The minimizer solves the equation
$$
\Delta U(x',x_n) = \chi_{\{V>0\}}(x') + \chi_{\{V=0\}}(x') [\partial_\nu U (x',0) + \partial_\nu U (x',1)],
$$
where $\partial_\nu U$ is the exterior normal derivative of $U$.
Several
regularity results are proven. It is shown
that the comparison principle does not hold for minimizers, which makes numerical approximation we present somehow challenging. |
|