Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy

TL;DR

This paper addresses the challenging problem of identifying a degenerate diffusion coefficient in a strongly degenerate parabolic PDE using optimal control techniques, providing existence, regularity, and explicit solutions.

Contribution

It introduces a novel approach for coefficient identification in degenerate parabolic equations, including existence, optimality conditions, and explicit solution characterization.

Findings

Existence of a control function in W^{1,∞} for the degenerate PDE.

Characterization of optimality conditions for the identification problem.

Explicit form and uniqueness of the identified diffusion coefficient.

Abstract

We study two identification problems in relation with a strongly degenerate parabolic diffusion equation characterized by a vanishing diffusion coefficient $u\in W^{1,\infty},$ with the property $\frac{1}{u}\notin L^{1}. $ The aim is to identify $u$ from certain observations on the solution, by a technique of nonlinear optimal control with control in coefficients. The existence of a controller $u$ which is searched in $% W^{1,\infty}$ and the determination of the optimality conditions are given for homogeneous Dirichlet boundary conditions. An approximating problem further introduced allows a better characterization of the optimality conditions, due to the supplementary regularity of the approximating state and dual functions and to a convergence result. Finally, an identification problem with final time observation and homogeneous Dirichlet-Neumann boundary conditions in the state system is considered. By using more technical arguments we provide the explicit form of $u$ and its uniqueness.