Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplace problem. The coefficient of the p-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the p-Laplacian, we use a regularization, sometimes referred to as the ε-p-Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted ε-p- Laplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by k. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each (ε, k)-level as the parameters tend to zero and infinity, respectively.This author’s research was supported by the DFG-EC315 “Engineering of Advanced Materials” and by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators