| Abstract: |
| Consider the following optimal minimization problem in the cylindrical domain $\Omega=D\times(0,1)$:
$$
\min_{\bar{\mathcal{R}}_\beta^D} \Phi(f)
$$
where
$$
\bar{\mathcal{R}}^D_\beta = \left\{f(x)\in L^\infty(\Omega)\colon f(x`,x_n) = f(x`),\,\, 0\leq f \leq 1,\,\,\int_D fdx = \beta \right\},
$$
$u_f\in W^{1,2}_0(\Omega)$ is the unique solution of $\Delta u_f=0$, and
$\Phi(f)=\int_\Omega |\nabla u_f|^2dx$.
We show the existence of the unique minimizer. Moreover, we show that for a particular $\alpha>0$ the function $U=\alpha-u_f$ minimizes the functional with nonlocal obstacle acting on function $V(x`)=\int_0^1 U(x`, t) dt $
$$
\int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x`)^+\,dx`,
$$
and 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 further regularity results are proven. It is shown that the comparison principle does not hold for minimizers, which makes numerical approximation we developed in [LM] somewhat challenging.
$$ $$
Keywords: rearrangement problems, free boundary, nonlocal obstacle
$$ $$
MSC Classification: 35R11, 35J60, 35R35, 35B51, 49J20, 65N06
$$ $$
[CM] Chipot, Michel; Mikayelyan, Hayk,
$\textit{On some nonlocal problems in the calculus of variations}$,
Nonlinear Anal. 217 (2022), Paper No. 112754, 17 pp.
$$ $$
[LM] Li, Zhilin; Mikayelyan, Hayk,
$\textit{Numerical analysis of a free boundary problem with non-local obstacles}$,
Appl. Math. Lett. 135 (2023), Paper No. 108414, 6 pp.
$$ $$
[M] Mikayelyan, Hayk,
$\textit{Cylindrical optimal rearrangement problem leading to a new type obstacle problem}$,
2018, ESAIM - Control, Optimisation and Calculus of Variations. 24, 2, p. 859-872 14 p. |
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