Corrigendum : An inverse problem in corrosion detection:stability estimates, J. Inv. Ill-posed Problems 12 (4) (2004), 349-367

TL;DR

This paper addresses an inverse problem related to corrosion detection, focusing on stability estimates to improve the understanding and reliability of corrosion assessment methods.

Contribution

It provides corrected and refined stability estimates for an inverse corrosion detection problem, enhancing the theoretical foundation of the method.

Findings

Refined stability estimates for corrosion inverse problems

Improved understanding of problem ill-posedness

Enhanced theoretical framework for corrosion detection

Abstract

Corrigendum : An inverse problem in corrosion detection:stability estimates, J. Inv. Ill-posed Problems 12 (4) (2004), 349-367.

Equations

$$\left\{\begin{array}[]{ll}\Delta u=&\mbox{in}\;\Omega,\\ \partial_{\nu}u+qu=0&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}u=f&\mbox{on}\;\Gamma_{2},\end{array}\right.$$

$$u=1+\ln r$$

$$u_k=u+2^{-k}{k}^{-2}(r^k+r^{-k})\cos (k\theta ),k\ge 1.$$

$$q=\frac{2}{1-\ln 2},$$

$$q=q_k=\frac{2+k^{-1}(2^{-2k+1}-2)\sin (k\theta )}{1-\ln 2+k^{-2}(2^{-2n}+1)\sin (k\theta )},k\ge 1,$$

$$L_K^p(D)=\{h\in L^p(D);\text{mbox}supp(h)\subset K\},1\le p\le \infty .$$

$$\left\{\begin{array}[]{ll}\Delta w=0&\mbox{in}\;\Omega,\\ w=0&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}w=f&\mbox{on}\;\Gamma_{2}.\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\Delta u=0&\mbox{in}\;\Omega,\\ \partial_{\nu}u+q_{0}u=0&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}u=f&\mbox{on}\;\Gamma_{2}.\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\Delta y=0&\mbox{in}\;\Omega,\\ \partial_{\nu}y+q_{0}y=\varphi_{1}&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}y=\varphi_{2}&\mbox{on}\;\Gamma_{2}.\end{array}\right.$$

$${\int }_{\Omega }|\nabla y{|}^{2}dx+{\int }_{{\Gamma }_1}{q}_0{y}^{2}d\sigma$$

$$\le \Vert ({\phi }_1,{\phi }_2){\Vert }_{L^2({\Gamma }_1)\oplus L^2({\Gamma }_2)}\Vert y{\Vert }_{H^1(\Omega )}.$$

$$h\to {\left({\int }_{\Omega }|\nabla h{|}^{2}dx+{\int }_{{\Gamma }_1}{q}_0{h}^{2}d\sigma \right)}^{1/2}$$

$$\Vert y{\Vert }_{H^1(\Omega )}\le {\kappa }_0\Vert ({\phi }_1,{\phi }_2){\Vert }_{L^2({\Gamma }_1)\oplus L^2({\Gamma }_2)},$$

$$\left\{\begin{array}[]{ll}\Delta y=0&\mbox{in}\;\Omega,\\ \partial_{\nu}y+y=(1-q_{0})y+\varphi_{1}&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}y=\varphi_{2}&\mbox{on}\;\Gamma_{2},\end{array}\right.$$

$$\Vert y{\Vert }_{H^{3/2}(\Omega )}\le {\kappa }_1\Vert ({\phi }_1,{\phi }_2){\Vert }_{L^2({\Gamma }_1)\oplus L^2({\Gamma }_2)}.$$

$$\Vert L\Vert :=\Vert L{\Vert }_{\mathcal{B}(L^2({\Gamma }_1)\oplus L^2({\Gamma }_2),H^{3/2}(\Omega ))}\le {\kappa }_1.$$

$$H_q:H^{3/2}(\Omega )\to H^{3/2}(\Omega ):H_q(u)=L\left(-q{u}_{|{\Gamma }_1},0\right).$$

$$h\in H^{3/2}(\Omega )\to u_{|{\Gamma }_1}\in C({\Gamma }_1),$$

$$\Vert H_q{\Vert }_{\mathcal{B}(H^{3/2}(\Omega ))}\le \kappa \Vert L\Vert \Vert q{\Vert }_{L^2({\Gamma }_1)}.$$

$$\Vert {\left(I-H_q\right)}^{-1}{\Vert }_{\mathcal{B}(H^{3/2}(\Omega ))}\le 2,q\in \mathcal{U}.$$

$$u_q({\phi }_1,{\phi }_2)={\left(I-H_q\right)}^{-1}L({\phi }_1,{\phi }_2).$$

$$u_q({\phi }_1,{\phi }_2)=L\left(-q{u}_{|{\Gamma }_1}+{\phi }_1,{\phi }_2\right),$$

$$\left\{\begin{array}[]{ll}\Delta u=0&\mbox{in}\;\Omega,\\ \partial_{\nu}u+(q_{0}+q)u=\varphi_{1}&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}u=\varphi_{2}&\mbox{on}\;\Gamma_{2}.\end{array}\right.$$

$$\Vert u_q({\phi }_1,{\phi }_2){\Vert }_{H^{3/2}(\Omega )}\le 2{\kappa }_1\Vert ({\phi }_1,{\phi }_2){\Vert }_{L^2({\Gamma }_1)\oplus L^2({\Gamma }_2)}.$$

$$\left\{\begin{array}[]{ll}\Delta u=0&\mbox{in}\;\Omega,\\ \partial_{\nu}u+(q_{0}+q)u=0&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}u=f&\mbox{on}\;\Gamma_{2}.\end{array}\right.$$

$$\Vert u_q{\Vert }_{H^{3/2}(\Omega )}\le 2{\kappa }_1\Vert f{\Vert }_{L^2({\Gamma }_2)}.$$

$$\Phi :q\in \mathcal{U}\cap L_{\overline{{\gamma }_1}}^2({\Gamma }_1)\to {\chi }_{{\Gamma }_1}\left[{\partial }_{\nu }{u}_q{}_{|{\gamma }_1}\right]\in L_{\overline{{\gamma }_1}}^2({\Gamma }_1)$$

$$\left\{\begin{array}[]{ll}\Delta v=0&\mbox{in}\;\Omega,\\ \partial_{\nu}v+q_{0}v=-p&\mbox{on}\;\Gamma_{1},\\ \partial_{\nu}v=0&\mbox{on}\;\Gamma_{2}.\end{array}\right.$$

$$\Vert q_1-q_2{\Vert }_{L^2({\Gamma }_1)}\le 2\Vert N^{-1}\Vert \Vert {\partial }_{\nu }{u}_{q_1}-{\partial }_{\nu }{u}_{q_2}{\Vert }_{L^2({\gamma }_1)},q_1,q_2\in \tilde{U}\cap L_{\overline{{\gamma }_1}}^2({\Gamma }_1).$$

$$B_{s,r}({\mathbb{R}}^k):=\{w\in {\mathcal{S}}^{\prime }({\mathbb{R}}^k);(1+|\xi {|}^2{)}^{s/2}\hat{w}\in L^r({\mathbb{R}}^k)\},$$

$$\Vert w{\Vert }_{B_{s,r}({\mathbb{R}}^k)}:={\Vert (1+|\xi {|}^2{)}^{s/2}\hat{w}\Vert }_{L^r({\mathbb{R}}^k)},$$

$$\Vert u_q{\Vert }_{H^3(\Omega )}\le C_0.$$

Full text

[TABLE]

we find a constant $C>0$, depending on $f$, $m$, $\Omega$ and $\gamma_{i}$, $i=1,2$, so that

[TABLE]

We now discuss briefly the stability around an arbitrary $q_{0}$. Let then $q_{0}\in L^{\infty}(\Gamma_{1})$ be non negative and non identically equal to zero and let $f\in L^{2}(\Gamma_{2})$ be non identically equal to zero. Denote by $u_{0}\in H^{3/2}(\Omega)$ the solution of the BVP

[TABLE]

As it is observed in [2],

[TABLE]

is an open dense subset of $\Gamma_{1}$.

Slight modifications of the preceding analysis allow us to prove the following result

[TABLE]

we find a constant $C>0$, depending on $f$, $m$, $\Omega$, $K$ and $\gamma_{2}$, so that

[TABLE]

Observe that, as in [2], the last theorem can be extended to the case where $\partial\Gamma_{1}\cap\partial\Gamma_{2}\neq\emptyset$.

In the most general case, in dimensions two and three, we can prove a stability estimate of triple logarithmic type (see [3, Theorem 4.9]).