Cylindrical optimal rearrangement problem leading to a new type obstacle problem

Mikayelyan, Hayk

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 859-872.

An optimal rearrangement problem in a cylindrical domain $\Omega =D\times (0,1)$ is considered, under the constraint that the force function does not depend on the $x_n$ variable of the cylindrical axis. This leads to a new type of obstacle problem in the cylindrical domain

$$\begin{array}{c}{\Delta }_u(x\text{'},x_n)={\chi }_{\{v>0\}}(x\text{'})+{\chi }_{\{v=0\}}(x\text{'})\left[{\partial }_{\nu }u(x\text{'},0)+{\partial }_{\nu }u(x\text{'},1)\right]\hfill \end{array}l$$

arising from minimization of the functional

$$\begin{array}{c}\hfill {\int }_{\Omega }\frac{1}{2}{\left|\nabla u\left(x\right)\right|}^2+{\chi }_{\{v>0\}}\left(x^{\text{'}}\right)u\left(x\right)\mathrm{d}x,\end{array}$$

where $v(x\text{'})={\int }_0^{1}u(x\text{'},t)\mathrm{d}t,\mathrm{and}$${\partial }_{\nu }u$ is the exterior normal derivative of $u$ at the boundary. Several existence and regularity results are proven and it is shown that the comparison principle does not hold for minimizers.

[1] I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem. Indiana Univ. Math. J. 50 (2001) 1077–1112. | DOI | MR | Zbl

[2] G.R. Burton, Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276 (1987) 225–253. | DOI | MR | Zbl

[3] G.R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré Anal. Non Linéaire 6 (1989) 295–319. | DOI | Numdam | MR | Zbl

[4] G.R. Burton and J.B. Mcleod, Maximisation and minimisation on classes of rearrangements. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991) 287–300. | DOI | MR | Zbl

[5] G.R. Burton and E.P. Ryan, On reachable sets and extremal rearrangements of control functions. SIAM J. Control Optimiz. 26 (1988) 1481–1489. | DOI | MR | Zbl

[6] L.A. Caffarelli, The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998) 383–402. | DOI | MR | Zbl

[7] F.H. Clarke, Yu. S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory, vol. 178 of Graduate Texts in Mathematics. Springer Verlag, New York (1998). | MR | Zbl

[8] B. Emamizadeh and Y. Liu, Constrained and unconstrained rearrangement minimization problems related to the p-Laplace operator. Israel J. Math. 206 (2015) 281–298 | DOI | MR | Zbl

[9] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001) | MR | Zbl

[10] B. Kawohl, Rearrangements and convexity of level sets in PDE, vol. 1150 of Lect. Notes Math. Springer Verlag, Berlin (1985) | DOI | MR | Zbl

[11] E.H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics. AMS, Providence, RI, 2nd edition (2001) | DOI | MR | Zbl