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Regularity for solutions of the total variation denoising problem. (English) Zbl 1228.94005
The authors study the local regularity properties of a local minimizer of the functional, $\int_\Omega|Du|+ {\lambda\over 2} \int_\Omega|u(x)- f(x)|^2\,dx,$ where $$\Omega$$ is an open set in $$\mathbb{R}^N$$, $$\lambda> 0$$, and $$f: \omega\to\mathbb{R}$$ is locally Hölder continuous. The purpose of this paper is to prove that $$u$$ is also locally Hölder continuous (with the same exponent). In addition is to prove a local Hölder regularity result for the solutions of the total variation based denoising problem assuming that the datum is locally Hölder continuous. The authors also prove a global estimate on the modulus of continuity of the solution in convex domain of $$\mathbb{R}^N$$ and some extensions of this result for the total variation minimization flow.

##### MSC:
 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 35J20 Variational methods for second-order elliptic equations 35J70 Degenerate elliptic equations 49N60 Regularity of solutions in optimal control
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##### References:
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