Jean Carlos Nakasato

Starting from the ‐Laplace thermistor model, in this paper we rigorously derive the effective model describing the current flow in a thin domain. We assume that the exponent is constant by parts and exhibiting the oscillating interface. Depending on the parameter describing these oscillations, we investigate the asymptotic behavior of the solution by using the adaptation of the periodic unfolding method. Our work is motivated by the industrial applications consisting of thin organic light‐emitting diodes.

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of this method to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We also compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.

In this work we analyse the asymptotic behaviour of the solutions of the p-Laplacian equation with homogeneous Neumann boundary conditions posed in bounded thin domains as R ε = ( x , y ) ∈ R 2 : x ∈ ( 0 , 1 ) and 0 < y < ε G x , x / ε α for some α > 0. We take a smooth function G : ( 0 , 1 ) × R ↦ R , L-periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter ɛ goes to zero and we determine the limit regime for three case: α < 1, α = 1 and α > 1.