The constrained least gradient problem in 𝑅ⁿ

Sternberg, Peter; Williams, Graham; Ziemer, William P.

doi:10.1090/S0002-9947-1993-1126213-2

The constrained least gradient problem in $\textbf {R}^ n$
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by Peter Sternberg, Graham Williams and William P. Ziemer PDF

Trans. Amer. Math. Soc. 339 (1993), 403-432 Request permission

Abstract:

We consider the constrained least gradient problem \[ \inf \left \{ {\int _\Omega {|\nabla u|dx:u \in {C^{0,1}}(\bar \Omega ),\quad |\nabla u| \leq 1\;{\text {a.e.}},u = g\;{\text {on}}\;\partial \Omega } } \right \}\] which arises as the relaxation of a nonconvex problem in optimal design. We establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem. We also establish the uniqueness of solutions and discuss their structure.

References

E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 250205, DOI 10.1007/BF01404309
Patricia Bauman and Daniel Phillips, A nonconvex variational problem related to change of phase, Appl. Math. Optim. 21 (1990), no. 2, 113–138. MR 1019397, DOI 10.1007/BF01445160
Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155–184. MR 454350, DOI 10.1007/BF02392236
L. A. Caffarelli and N. M. Rivière, Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 2, 289–310. MR 412940
Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. MR 110078, DOI 10.1090/S0002-9947-1959-0110078-1
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Wendell H. Fleming and Raymond Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. MR 114892, DOI 10.1007/BF01236935
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
Marvin J. Greenberg and John R. Harper, Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. MR 643101
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
Robert Kohn and Gilbert Strang, The constrained least gradient problem, Nonclassical continuum mechanics (Durham, 1986) London Math. Soc. Lecture Note Ser., vol. 122, Cambridge Univ. Press, Cambridge, 1987, pp. 226–243. MR 926504, DOI 10.1017/CBO9780511662911.015

Fibered structures in optimal design

Robert V. Kohn and Gilbert Strang, Optimal design and relaxation of variational problems. I, Comm. Pure Appl. Math. 39 (1986), no. 1, 113–137. MR 820342, DOI 10.1002/cpa.3160390107
Umberto Massari and Mario Miranda, Minimal surfaces of codimension one, North-Holland Mathematics Studies, vol. 91, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 95. MR 795963
Harold R. Parks and William P. Ziemer, Jacobi fields and regularity of functions of least gradient, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 505–527. MR 808421

Theory of the integral

Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
Leon Simon, A strict maximum principle for area minimizing hypersurfaces, J. Differential Geom. 26 (1987), no. 2, 327–335. MR 906394
Peter Sternberg, Graham Williams, and William P. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math. 430 (1992), 35–60. MR 1172906
Peter Sternberg, William P. Ziemer, and Graham Williams, $C^{1,1}$-regularity of constrained area minimizing hypersurfaces, J. Differential Equations 94 (1991), no. 1, 83–94. MR 1133542, DOI 10.1016/0022-0396(91)90104-H
Italo Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math. 334 (1982), 27–39. MR 667448, DOI 10.1515/crll.1982.334.27
Italo Tamanini, Il problema della capillarità su domini non regolari, Rend. Sem. Mat. Univ. Padova 56 (1976), 169–191 (1978) (Italian). MR 483992
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3