Geodesic
Solution of Dirichlet's problem for the equation $\Delta u=-1$ in a convex region
Matematičeskie zametki, Tome 9 (1971) no. 1, pp. 89-92

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Let $u$ be a solution of the following boundary-value problem: $u|_\Gamma=0$, where $\Gamma$ is a closed convex curve and $\Delta u=-1$ in the region $D$ bounded by $\Gamma$. Then $u$ has only one local maximum, and all its level curves are convex. 
@article{MZM_1971_9_1_a12,
     author = {L. G. Makar-Limanov},
     title = {Solution of {Dirichlet's} problem for the equation $\Delta u=-1$ in a convex region},
     journal = {Matemati\v{c}eskie zametki},
     pages = {89--92},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_1_a12/}
}
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L. G. Makar-Limanov. Solution of Dirichlet's problem for the equation $\Delta u=-1$ in a convex region. Matematičeskie zametki, Tome 9 (1971) no. 1, pp. 89-92. http://geodesic.mathdoc.fr/item/MZM_1971_9_1_a12/