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Big pieces of $$C^{1,\alpha}$$-graphs for minimizers of the Mumford-Shah functional. (English) Zbl 0960.49024
Summary: We consider the generalization of the Mumford-Shah functional defined by $J(u, K)= \int_{\Omega\setminus K}|u-g|^2+ \int_{\Omega\setminus K}|\nabla u|^2+ H^{n- 1}(K),$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^n$$ $$(n\geq 2)$$, $$g$$ a bounded measurable function on $$\Omega$$, $$K$$ a relatively closed subset of $$\Omega$$, $$H^{n-1}(K)$$ denotes the $$(n-1)$$-dimensional Hausdorff measure on $$K$$ and $$u\in W^{1,2}(\Omega\setminus K)$$. We prove here that there exist $$\alpha\in (0,1)$$ and $$C>1$$ such that if $$(u,K)$$ is an irreducible minimizer for $$J$$ and $$B(x,r)$$ a ball centered on $$K$$, contained in $$\Omega$$, with radius $$r\leq 1$$, then there is a ball $$B$$ centered on $$K$$, contained in $$B(x,r)$$, with radius $$\geq C^{-1}r$$, such that $$K\cap B$$ is a $$C^{1,\alpha}$$-hypersurface. Moreover, the constants $$\alpha$$, $$C$$ and the $$C^{1,\alpha}$$-constant for $$K\cap B$$ depend only on $$n$$ and $$\|g\|_\infty$$. In particular, the Hausdorff dimension of the set of points in $$K$$ around which $$K$$ is a $$C^{1,\alpha}$$-hypersurface is strictly less than $$n-1$$.

MSC:
 49N60 Regularity of solutions in optimal control 49J10 Existence theories for free problems in two or more independent variables 49Q20 Variational problems in a geometric measure-theoretic setting
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References:
 [1] ’ R. Adams , ” Sobolev Spaces ”, Academic Press , New York - London , 1975 . MR 450957 | Zbl 0314.46030 · Zbl 0314.46030 [2] L. Ambrosio , Existence theory for a new class of variational problems , Arch. Rational Mech. Anal. 111 ( 1990 ), 291 - 322 . MR 1068374 | Zbl 0711.49064 · Zbl 0711.49064 [3] L. Ambrosio - N. Fusco - D. Pallara , Partial regularity of free discontinuity sets II , Ann. Scuola Norm. Sup. Pisa Cl. Sci. ( 4 ) 24 ( 1997 ), 39 - 62 . Numdam | MR 1475772 | Zbl 0896.49024 · Zbl 0896.49024 [4] M. Carriero - A. Leaci , Existence theorem for a Dirichlet problem with free discontinuity set , Nonlinear Anal. 15 ( 1990 ), 661 - 677 . MR 1073957 | Zbl 0713.49003 · Zbl 0713.49003 [5] G. Dal Maso - J.-M. Morel - S. Solimini , A variational method in image segmentation: existence and approximation results , Acta Math. 168 ( 1992 ), 89 - 151 . MR 1149865 | Zbl 0772.49006 · Zbl 0772.49006 [6] G. David , ” Wavelets and singular integrals on curves and surfaces ”, Lecture Notes in Math , Vol. 1465 , Springer-Verlag , Berlin , 1991 . MR 1123480 | Zbl 0764.42019 · Zbl 0764.42019 [7] G. David , C1-arcs for minimizers of the Mumford-Shah functional ,, SIAM J. Appl. Math 56 ( 1996 ), 783 - 888 . MR 1389754 | Zbl 0870.49020 · Zbl 0870.49020 [8] G. David - S. Semmes , ” Analysis of and on uniformly rectifiable sets ”, Math. Surveys Monogr. , Vol. 38 , Amer. Math. Soc. , Providence , 1993 . MR 1251061 | Zbl 0832.42008 · Zbl 0832.42008 [9] G. David - S. Semmes , On the singular sets of minimizers of the Mumford-Shahfunctional , J. Math. Pures Appl. ( 4 ) 75 ( 1996 ), 299 - 342 . MR 1411155 | Zbl 0853.49010 · Zbl 0853.49010 [10] G. David - S. Semmes , Uniform rectifiability and singular sets , Ann. Inst. H. Poincaré Anal. Non Linéaire 13 ( 1996 ), 383 - 443 . Numdam | MR 1404317 | Zbl 0908.49030 · Zbl 0908.49030 [11] E. De Giorgi - M. Carriero - A. Leaci , Existence theorem fora minimum problem with free discontinuity set , Arch. Rational Mech. Anal. 108 ( 1989 ), 195 - 218 . MR 1012174 | Zbl 0682.49002 · Zbl 0682.49002 [12] F. Maddalena - S. Solimini , Regularity properties of free discontinuity sets , preprint. · Zbl 1024.49013 [13] D. Mumford - J. Shah , Optimal approximations by piecewise smooth functions and associated variational problems , Comm. Pure Appl. Math. 42 ( 1989 ), 577 - 685 . MR 997568 | Zbl 0691.49036 · Zbl 0691.49036
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