Fréchet differentiability of the solution of the heat equation with respect to a nonlinear boundary condition
type: Item
instance of: scholarly article
title: Fréchet differentiability of the solution of the heat equation with respect to a nonlinear boundary condition
zbMATH Open document ID: 0858.35012
DOI: 10.4171/ZAA/718
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1996-10-14T00:00:00Z
review text: The paper concerns the heat equation in a smooth bounded domain \(\Omega\subset\mathbb{R}^N\) with nonlinear boundary condition \(\partial_nu=\alpha(u) (\theta-u)\) and initial condition \(u=u_0\). Here \(\theta\in\mathbb{R}\) is fixed, and the nonlinear function \(\alpha\) is defined on the closed interval \([\theta_1,\theta_2]\) with \[ \theta_1:=\min\{\theta,\;\inf_{x\in\Omega} u_0(x)\}\qquad \text{and} \qquad \theta_2:=\max\{\theta,\;\sup_{x\in\Omega} u_0(x)\} \] and positive valued. The author shows that the map \(\alpha\mapsto u|_{[0,T]\times \partial\Omega}\) is Fréchet differentiable everywhere (as a map from \(C^1[\theta_1,\theta_2]\) into \(C([0,T]\times \partial\Omega)\), resp. in certain points (as a map from \(C^{0,1}[\theta_1,\theta_2]\) into \(C([0,T]\times \partial\Omega)\). The maximum principle as well as analytic semigroup and boundary integral equation techniques are used.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 935174
zbMATH Keywords: maximum principle
zbMATH Keywords: analytic semigroup
zbMATH Keywords: boundary integral equation
author: Arnd Rösch
MaRDI profile type: Publication
cites work: Q3751968
cites work: Parabolic evolution equations and nonlinear boundary conditions
cites work: The superposition operator in function spaces - A survey
cites work: Q4040886
cites work: Optimal control for parabolic equations
cites work: On NEMYTSKIJ Operators inLp-Spaces of Abstract Functions
cites work: Q3244654
cites work: Q4188332
cites work: Q3933509
cites work: Q4056239
cites work: Identification of nonlinear heat transfer laws by optimal control
cites work: Q3139931
cites work: Q3336199
cites work: Q3713040
cites work: Q4765632
full work available at URL: https://doi.org/10.4171/zaa/718
OpenAlex ID: W1963621439
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description: scientific article; zbMATH DE number 935174