This paper concerns minimization and maximization of the first eigenvalue in problems involving the p-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of N = 2 and p close to 2, our equation models the vibration of a nonhomogeneous membrane Ω which is fixed along the boundary. Given several materials (with different densities) of total extension |Ω|, we investigate the location of these material inside Ω so as to minimize or maximize the first mode in the vibration of the membrane.

In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution towhere λ ± > 0 are given constants and μ ± , are non-negative bounded Radon measures, with compact support. Our primary concern is to discuss existence and geometric properties of solutions.

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Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation This paper concerns optimization problems related to bi-harmonic equations subject to either Navier or Dirichlet homogeneous boundary conditions. Physically, in dimension two, our equation models the deformation of an elastic plate which is either hinged or clamped along the boundary, under load. We discuss existence, uniqueness, and properties of the optimizers.

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