On a degenerate problem in the calculus of variations
HTML articles powered by AMS MathViewer
- by Guy Bouchitté and Pierre Bousquet PDF
- Trans. Amer. Math. Soc. 371 (2019), 777-807 Request permission
Abstract:
We establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations. The proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets. We exploit the uniqueness and the regularity results to explore some of their qualitative properties. In particular, we emphasize the link between the supports of the solutions and the Cheeger problem.References
- Giovanni Alberti, Stefano Bianchini, and Gianluca Crippa, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 4, 863–902. MR 3184572
- Jean Jacques Alibert, Guy Bouchitté, Ilaria Fragalà, and Ilaria Lucardesi, A nonstandard free boundary problem arising in the shape optimization of thin torsion rods, Interfaces Free Bound. 15 (2013), no. 1, 95–119. MR 3062575, DOI 10.4171/IFB/296
- F. Alter, V. Caselles, and A. Chambolle, A characterization of convex calibrable sets in $\Bbb R^N$, Math. Ann. 332 (2005), no. 2, 329–366. MR 2178065, DOI 10.1007/s00208-004-0628-9
- François Alter and Vicent Caselles, Uniqueness of the Cheeger set of a convex body, Nonlinear Anal. 70 (2009), no. 1, 32–44. MR 2468216, DOI 10.1016/j.na.2007.11.032
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Gabriele Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318 (1984). MR 750538, DOI 10.1007/BF01781073
- Guy Bouchitté, Ilaria Fragalà, Ilaria Lucardesi, and Pierre Seppecher, Optimal thin torsion rods and Cheeger sets, SIAM J. Math. Anal. 44 (2012), no. 1, 483–512. MR 2888297, DOI 10.1137/110828538
- L. Brasco, Global $L^\infty$ gradient estimates for solutions to a certain degenerate elliptic equation, Nonlinear Anal. 74 (2011), no. 2, 516–531. MR 2733227, DOI 10.1016/j.na.2010.09.006
- V. Caselles, A. Chambolle, and M. Novaga, Some remarks on uniqueness and regularity of Cheeger sets, Rend. Semin. Mat. Univ. Padova 123 (2010), 191–201. MR 2683297, DOI 10.4171/RSMUP/123-9
- A. Cellina, Uniqueness and comparison results for functionals depending on $\nabla u$ and on $u$, SIAM J. Optim. 18 (2007), no. 3, 711–716. MR 2345964, DOI 10.1137/060657455
- Maria Colombo and Alessio Figalli, Regularity results for very degenerate elliptic equations, J. Math. Pures Appl. (9) 101 (2014), no. 1, 94–117 (English, with English and French summaries). MR 3133426, DOI 10.1016/j.matpur.2013.05.005
- Ivar Ekeland and Roger Témam, Convex analysis and variational problems, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Translated from the French. MR 1727362, DOI 10.1137/1.9781611971088
- Irene Fonseca, Nicola Fusco, and Paolo Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM Control Optim. Calc. Var. 7 (2002), 69–95. MR 1925022, DOI 10.1051/cocv:2002004
- Mariano Giaquinta, Giuseppe Modica, and Jiří Souček, Cartesian currents in the calculus of variations. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 37, Springer-Verlag, Berlin, 1998. Cartesian currents. MR 1645086, DOI 10.1007/978-3-662-06218-0
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
- Enrico Giusti, On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions, Invent. Math. 46 (1978), no. 2, 111–137. MR 487722, DOI 10.1007/BF01393250
- 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
- Enrico Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933, DOI 10.1142/9789812795557
- Robert L. Jerrard, Amir Moradifam, and Adrian I. Nachman, Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math. 734 (2018), 71–97. MR 3739314, DOI 10.1515/crelle-2014-0151
- 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
- Robert V. Kohn and Gilbert Strang, Optimal design and relaxation of variational problems. II, Comm. Pure Appl. Math. 39 (1986), no. 2, 139–182. MR 820067, DOI 10.1002/cpa.3160390202
- Robert V. Kohn and Gilbert Strang, Optimal design and relaxation of variational problems. III, Comm. Pure Appl. Math. 39 (1986), no. 3, 353–377. MR 829845, DOI 10.1002/cpa.3160390305
- Luca Lussardi and Elvira Mascolo, A uniqueness result for a class of non-strictly convex variational problems, J. Math. Anal. Appl. 446 (2017), no. 2, 1687–1694. MR 3563054, DOI 10.1016/j.jmaa.2016.09.060
- Paolo Marcellini, A relation between existence of minima for nonconvex integrals and uniqueness for nonstrictly convex integrals of the calculus of variations, Mathematical theories of optimization (Genova, 1981) Lecture Notes in Math., vol. 979, Springer, Berlin-New York, 1983, pp. 216–231. MR 713812
- P. Marcellini, Some remarks on uniqueness in the calculus of variations, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. IV (Paris, 1981/1982) Res. Notes in Math., vol. 84, Pitman, Boston, Mass.-London, 1983, pp. 148–153 (English, with French summary). MR 716516
- Umberto Massari, Esistenza e regolarità delle ipersuperfice di curvatura media assegnata in $R^{n}$, Arch. Rational Mech. Anal. 55 (1974), 357–382 (Italian). MR 355766, DOI 10.1007/BF00250439
- Frank Morgan, Geometric measure theory, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR 2455580
- Terrence Napier and Mohan Ramachandran, An introduction to Riemann surfaces, Cornerstones, Birkhäuser/Springer, New York, 2011. MR 3014916
- Enea Parini, An introduction to the Cheeger problem, Surv. Math. Appl. 6 (2011), 9–21. MR 2832554
- Jean-Pierre Raymond, An anti-plane shear problem, J. Elasticity 33 (1993), no. 3, 213–231. MR 1266980, DOI 10.1007/BF00043249
- 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, Graham Williams, and William P. Ziemer, The constrained least gradient problem in $\textbf {R}^n$, Trans. Amer. Math. Soc. 339 (1993), no. 1, 403–432. MR 1126213, DOI 10.1090/S0002-9947-1993-1126213-2
- 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
Additional Information
- Guy Bouchitté
- Affiliation: IMATH, EA 2134, Université du Sud Toulon-Var, BP 20132 - 83957 La Garde Cedex, France
- Email: bouchitte@univ-tln.fr
- Pierre Bousquet
- Affiliation: Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, F-31062 Toulouse Cedex 9, France
- MR Author ID: 791850
- Email: pierre.bousquet@math.univ-toulouse.fr
- Received by editor(s): November 30, 2016
- Published electronically: October 17, 2018
- Additional Notes: Part of this work was written during a visit of the second author to Toulon. The support of IMATH is kindly acknowledged.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 777-807
- MSC (2010): Primary 35A02, 49N60, 49J45
- DOI: https://doi.org/10.1090/tran/7570
- MathSciNet review: 3885161