Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem
type: Item
instance of: scholarly article
title: Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem
zbMATH Open document ID: 0879.49020
DOI: 10.4171/ZAA/723
author: Eduardo Casas
author: Fredi Tröltzsch
author: A. Unger
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1998-01-25T00:00:00Z
review text: Let \(\varphi:\mathbb{R}\to\mathbb{R}\), \(\psi,b:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) and all derivatives up to second-order be globally Lipschitz continuous, \(b(.,y)\) decreasing for all \(y\in\mathbb{R}\); \(\Omega\) a bounded domain of \(\mathbb{R}^n\) with a sufficiently smooth boundary such that \(W^{1,p}(\Omega)\) is continuously embedded in \(C(\overline\Omega)\) and \(w(u)\in W^{1,p}(\Omega)\) for some \(p>n\), where \(u\in U\equiv\{u\in L^\infty(\partial\Omega): m\leq u\leq M\}\) and \(m,M\in\mathbb{R}\), \[ \int_\Omega \Biggl(\sum^n_{i=1} \partial_iw(u)\partial_iv+ w(u)v\Biggr)dx= \int_{\partial\Omega} b(w(u),u)vd\sigma\quad\text{for all }v\in W^{1,2}(\Omega), \] \[ |w(u_1)- w(u_2)|_{W^{1,p}(\Omega)}\leq c_p|u_1- u_2|_{L^p(\partial\Omega)}\quad\text{for }c_p>0; \] \[ L:(C(\overline\Omega)\cap W^{1,2}(\Omega))\times U\times(C(\overline\Omega)\cap W^{1,2}(\Omega))\to \mathbb{R}, \] \[ L(w,u,v)= \int_\Omega\varphi\circ wdx+ \int_{\partial\Omega} \psi(w,u)d\sigma-\int_\Omega \Biggl(\sum^n_{i=1} \partial_iw\partial_iv+ wv\Biggr)dx+ \int_{\partial\Omega}b(w,u)vd\sigma. \] Theorem. Let \(u_0\in U\), \(v_0\in C(\overline\Omega)\cap W^{1,2}(\Omega)\) such that \[ \int_{\partial\Omega} (\partial_2\psi(w(u_0),u_0)+ \partial_2b(w(u_0),u_0)v_0)(u- u_0)d\sigma\geq 0\quad\text{for all }u\in U, \] \[ \begin{multlined}\int_\Omega \Biggl(\sum^n_{i=1}\partial_i v_0\partial_iv+ v_0v- \varphi'\circ w(u_0)v\Biggr)dx=\\ \int_{\partial\Omega}(\partial_1b(w(u_0),u_0)v_0+ \partial_1\psi(w(u_0),u_0))vd\sigma\text{ for all }v\in W^{1,2}(\Omega);\end{multlined} \] let \(\varepsilon>0\) and \[ \Gamma_\varepsilon= \{x\in\partial\Omega:|\partial_2 \psi(w(u_0)(x), u_0(x))+ \partial_2b(w(u_0)(x), u_0(x))v_0(x)|\geq \varepsilon\}; \] suppose that there exists \(\alpha>0\) such that \[ L(.,.,v_0)''(w(u_0),u_0)((h,u- u_0),(h,u- u_0))\geq\alpha|(h,u- u_0)|^2_{W^{1,2}(\Omega)\times L^2(\partial\Omega)}\text{ for }u\in U \] such that \(u(x)= u_0(x)\) for a.e. \(x\in\Gamma_\varepsilon\) an…
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35J99
Mathematics Subject Classification ID: 49K27
zbMATH DE Number: 935179
zbMATH Keywords: sufficient optimality conditions
zbMATH Keywords: nonlinear elliptic boundary control
reviewed by: Gianfranco Bottaro
MaRDI profile type: Publication
cites work: Q4315454
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full work available at URL: https://doi.org/10.4171/zaa/723
OpenAlex ID: W2036112861
description: scientific article; zbMATH DE number 935179