Uniform local energy estimates and partial regularity for space partitions

References

• Almgren , FJ Jr and Taylor , JE . 1996 . Soap Bubble Clusters . Forma , 11 ( 3 ) : 199 – 207 . and reprinted in a book in The Kelvin Problem, W. Denis, ed., Taylor and Francis, London, 1996, pp. 37–45
   Google Scholar
• Ambrosio , L , Fusco , N and Pallara , D . 2000 . Functions of Bounded Variation and Free Discontinuity Problems , New York : Oxford University Press .
   Google Scholar
• Bellettini , G , Caselles , V and Novaga , M . 2002 . The total variation flow in ℝ N . J. Differ. Equ. , 184 ( 2 ) : 475 – 525 .
   Web of Science ®Google Scholar
• Bellettini , G , Chermisi , M and Novaga , M . 2006 . Crystalline curvature flow of planar networks . Interfaces Free Bound. , 8 ( 4 ) : 481 – 521 .
   Web of Science ®Google Scholar
• Braides , A . 1998 . Approximation of Free-Discontinuity Problems , Berlin : Springer-Verlag .
   Google Scholar
• Burago , YD and Zalgaller , VA . 1988 . Geometric Inequalities , New York : Springer-Verlag .
   Google Scholar
• Caraballo , DG . 2008 . Crystals and polycrystals in ℝ n : Lower semicontinuity and existence . J. Geom. Anal. , 18 ( 1 ) : 68 – 88 .
   Web of Science ®Google Scholar
• Gurtin , ME , Williams , WO and Ziemer , WP . 1986 . Geometric measure theory and the axioms of continuum thermodynamics . Arch. Ration. Mech. Anal. , 92 ( 1 ) : 1 – 22 .
   Web of Science ®Google Scholar
• Morgan , F . 2008 . Geometric Measure Theory: A Beginner's Guide, , 4th , New York : Academic Press .
   Google Scholar
• Taylor , JE . 2003 . Some mathematical challenges in materials science . Bull. AMS , 40 ( 1 ) : 69 – 87 .
   Web of Science ®Google Scholar
• Evans , LC and Gariepy , RF . 1992 . Measure Theory and Fine Properties of Functions , Boca Raton , FL : CRC Press .
   Google Scholar
• Giusti , E . 1984 . Minimal Surfaces and Functions of Bounded Variation , Boston : Birkhäuser .
   Google Scholar
• Krantz , SG and Parks , HR . 1999 . The Geometry of Domains in Space , Boston : Birkhäuser .
   Google Scholar
• De Giorgi , E . 1960/1961 . Frontiere orientate di misura minima . Sem. Mat. Sc. Norm. Sup. Pisa , Editrice Tecnico Scientifica, 1961
   Google Scholar
• Reifenberg , ER . 1960 . Solution of the plateau problem for m-dimensional surfaces of varying topological type . Acta Math. , 104 : 1 – 92 .
   Web of Science ®Google Scholar
• Miranda , M . 1965 . Sul minimo dell'integrale del gradiente di una funzione . Ann. Sc. Norm. Sup. Pisa , 19 : 627 – 665 .
   Google Scholar
• Federer , H . 1969 . Geometric Measure Theory , New York : Springer-Verlag .
   Google Scholar
• Allard , W . 1972 . On the first variation of a varifold . Ann. Math. , 95 : 417 – 491 .
   Web of Science ®Google Scholar
• Massari , U . 1974 . Esistenza e regolaritè delle ipersuperfici di curvatura media assegnata in ℝ n . Arch. Ration. Mech. Anal. , 55 : 357 – 382 .
   Google Scholar
• Almgren , FJ Jr . 1976 . Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints . Mem. Amer. Math. Soc. , 165 : 1 – 199 .
   Google Scholar
• Bombieri , E . 1982 . Regularity theory for almost minimal currents . Arch. Ration. Mech. Anal. , 78 : 99 – 130 .
   Web of Science ®Google Scholar
• Tamanini , I . 1982 . Boundaries of Caccioppoli sets with Hölder continuous normal vector . J. Reine Angew. Math. , 334 : 27 – 39 .
   Web of Science ®Google Scholar
• Ambrosio , L and Paolini , E . 1998 . Partial regularity for quasi minimizers of perimeter . Ricerche di Matematica , 48 ( Suppl. ) : 167 – 186 .
   Google Scholar
• Caraballo , DG . 1997 . A variational scheme for the evolution of polycrystals by curvature , Ph.D. thesis Princeton University . (Thesis advisor: Frederick J. Almgren Jr)
   Google Scholar
• Mattila , P . 1995 . Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability , Cambridge : Cambridge University Press .
   Google Scholar
• Caraballo , DG . 2009 . The triangle inequalities and lower semi-continuity of surface energy of partitions . Proc. R. Soc. Edinb. , 139A : 449 – 457 .
   Google Scholar
• Almgren , FJ Jr , Taylor , JE and Wang , L . 1993 . Curvature driven flows: A variational approach . SIAM J. Control Optim. , 31 ( 2 ) : 387 – 438 .
   Web of Science ®Google Scholar
• Yip , NK . 2002 . “ Stochastic curvature driven flows ” . In in Stochastic Partial Differential Equations and Applications (Trento, 2002), Lecture Notes in Pure and Applied Mathematics , Vol. 227 , 443 – 460 . New York : Dekker .
   Google Scholar
• Caraballo , DG . 2005 . 2-Dimensional flat curvature flow of crystals . Interfaces Free Bound. , 7 ( 3 ) : 241 – 254 .
   Web of Science ®Google Scholar
• Caraballo , DG . B2-convexity implies strong and weak lower semicontinuity for partitions of Rn . Pacific Journal of Math. , (to appear)
   Web of Science ®Google Scholar
• Caraballo , DG . 2010 . B2-convexity and surface energy of space partitions . J. Mathé. Pures Appl. , 94 ( 1 ) : 58 – 67 .
   Web of Science ®Google Scholar