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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References:

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