Fréchet differentiability of the solution of the heat equation with respect to a nonlinear boundary condition
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.
scientific article; zbMATH DE number 935174
scientific article