On the stability of integral equations of the first kind with logarithmic kernels
Among other ones, the author discusses the problem \[ - \int_{\Gamma}\sigma (y)\log | x-y| dS_ y+\omega =f(x),\quad x\in \Gamma,\quad \int_{\Gamma}\sigma (y)dS_ y=b \] where \(\Gamma\) is a smooth, simple, closed curve on the plane, f is a given function and by is a given constant (both not known exactly, \(| b_{\epsilon}-b| <\epsilon\), \(\| f_{\epsilon}-f\|_{L^ 2(\Gamma)}\leq \epsilon)\). The function \(\sigma\) and the constant \(\omega\) are to be determined. The problem is discretized by means of Galerkin finite elements. The author derives error estimates on the norms of \(H^ t(\Gamma)\times {\mathbb{R}}\), -\(\leq t\leq m\). The results are consistent with those obtained by the Tikhonov regularization procedure.
scientific article; zbMATH DE number 3980434
scientific article