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Zeitschrift für Analysis und ihre Anwendungen


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Volume 15, Issue 3, 1996, pp. 603–618
DOI: 10.4171/ZAA/718

Published online: 1996-09-30

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

Arnd Rösch[1]

(1) Universität Duisburg-Essen, Germany

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.

Keywords: Fréchet differentiability, heat equation, nonlinear boundary conditions, identification of the heat function, inverse problems

Rösch Arnd: Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition. Z. Anal. Anwend. 15 (1996), 603-618. doi: 10.4171/ZAA/718