Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition

Abstract

We consider the heat equation $\frac{\partial u}{\partial t}(t,x)={\Delta }_{x}u(t,x)$ with a nonlinear function a in the boundary condition $\frac{\partial u}{\partial n}(t,x)=\alpha ((u(t,z))(\vartheta –u(t,x))$ depending on the boundary values $x$ of the solution u of the initial-boundary value problem only and belonging to a set of admissible differentiable or uniformly Lipschitz continuous functions. For this problem Lipschitz continuity and Fréchet differentiability of the mapping $\Phi :\alpha \mapsto x$ under different assumptions are derived.