JournalszaaVol. 15, No. 3pp. 603–618

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

  • Arnd Rösch

    Universität Duisburg-Essen, Germany
Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition cover

Abstract

We consider the heat equation ut(t,x)=Δxu(t,x)\frac{\partial u}{\partial t} (t, x) = \Delta _x u(t,x) with a nonlinear function a in the boundary condition un(t,x)=α((u(t,z))(ϑu(t,x))\frac{\partial u}{\partial n} (t, x) =\alpha ((u(t,z))(\vartheta – u(t,x)) depending on the boundary values xx 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 Φ:αx\Phi : \alpha \mapsto x under different assumptions are derived.

Cite this article

Arnd Rösch, Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition. Z. Anal. Anwend. 15 (1996), no. 3, pp. 603–618

DOI 10.4171/ZAA/718