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Search: All articles in the CJM digital archive with keyword shape optimization

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1. CJM Online first

Karpukhin, Mikhail A.
Steklov problem on differential forms
In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\Lambda$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\Lambda$ and prove a Hersch-Payne-Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\Lambda$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares a lot of important properties with the classical Steklov eigenvalue problem on surfaces.

Keywords:Dirichlet-to-Neumann map, differential form, Steklov eigenvalue, shape optimization
Categories:58J50, 58J32, 35P15

2. CJM 2011 (vol 64 pp. 1058)

Plakhov, Alexander
Optimal Roughening of Convex Bodies
A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface, and do not interact with each other. We consider a generalization of Newton's minimal resistance problem: given two bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2 \subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the resistance in the class of connected bodies $B$ such that $C_1 \subset B \subset C_2$. We prove that the infimum of resistance is zero; that is, there exist "almost perfectly streamlined" bodies.

Keywords:billiards, shape optimization, problems of minimal resistance, Newtonian aerodynamics, rough surface
Categories:37D50, 49Q10

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