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Singular free boundary problem from image processing. (English) Zbl 1081.35054

Consider the singular free boundary problem \[ \begin{aligned} u_t=\Phi(u_x)_x, &\qquad x\in(\xi^0(t),\xi^1(t)),\;t\in(0,T),\\ u(\xi^i(t),t)=i,\;\Phi(u_x)(\xi^i(t),t))=\Phi_\infty, &\qquad i=0,1,\;t\in(0,T),\\ u(x,0)=u_0(x)\in(0,1), &\qquad x\in(\ell,r), \\ \xi^0(0)=\ell,\;\xi^1(0)=r, &{} \end{aligned} \] where \(\Phi\) is strictly increasing and \(\Phi_\infty=\lim_{v\to+\infty}\Phi(v)<\infty\). This problem arises in the study of the technique of contour enhancement in image processing. Under suitable assumptions on \(\Phi\) and \(u_0\) (\(u_0\) need not be monotone), the authors prove the well-posedness for the singular problem and the convergence of the approximation by means of combustion-type free-boundary problems.

MSC:

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
68U10 Computing methodologies for image processing
35R35 Free boundary problems for PDEs
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