Abstract
For a bounded Lipschitz domain Ω ⊂ ℝn, n ≥ 2, and a function u O ∈ W 11 (Ω;ℝN) we consider the variational problem
where f:ℝnN → [0,∞]is a strictly convex integrand of linear growth, i.e.
holds with suitable constants a, A > 0, b,B ∈ ℝ.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ambrosio, L., Fusco, N., Pallara, D. (2000): Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, Clarendon Press, Oxford
Anzellotti, G., Giaquinta, M. (1988): Convex functional and partial regularity. Arch. Rat. Mech. Anal., 102. 243–272
Bildhauer, M. (2001): Convex Variational Problems with Linear, Nearly Linear and/or Anisotropic Growth Conditions. Habilitationsschrift (submitted), Saarland University, Saarbrücken
Bildhauer, M. (2002): A priori gradient estimates for bounded generalized solutions of a class of variational problems with linear growth. To appear in J. Convex Anal.
Bildhauer, M., Fuchs, M. (1999): Regularity for dual solutions and for weak cluster points of minimizing sequences of variational problems with linear growth. Zap. Nauchn. Sem. St.-Petersburg Odtel. Math. Inst. Steklov (POMI), 259, 46–66
Bildhauer, M., Fuchs, M. (2000): On a class of variational integrals with linear growth satisfying the condition of μ-ellipticity. Preprint Bonn University/SFB 256 no. 681
Bildhauer, M., Fuchs, M. (2002): Relaxation of convex variational problems with linear growth defined on classes of vector-valued functions. To appear in Algebra and Analiz 14
Bildhauer, M., Fuchs, M., Mingione, G. (2001): A priori gradient bounds and local C 1,α-estimates for (double) obstacle problems under nonstandard growth conditions. Z. Anal. Anw., 20, 959–985
Bombieri, E., DeGiorgi, E., Miranda, M. (1969): Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Rat. Mech. Anal., 32, 255–267
Giaquinta, M., Modica, G., Souček, J. (1979): Functionals with linear growth in the calculus of variations. Comm. Math. Univ. Carolinae, 20, 143–172
Giusti, E. (1984): Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics 80, Birkhäuser, Boston-Basel-Stuttgart
Ladyzhenskaya, O.A., Ural’tseva, N.N. (1964): Linear and Quasilinear Elliptic Equations. Nauka, Moskow (in Russian) English translation (1968): Academic Press, New York
Ladyzhenskaya, O.A., Ural’tseva, N.N. (1970): Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Comm. Pure Appl. Math., 23, 667–703
Massari, U., Miranda, M. (1984): Minimal Surfaces of Codimension One. North-Holland Mathematics Studies 91, North-Holland, Amsterdam
Miranda, M. (1967): Disequaglianze di Sobolev sulle ipersuperfici minimali. Rend. Sem. Mat. Univ. Padova, 38, 69–79
Reshetnyak, Y. (1968): Weak convergence of completely additive vector functions on a set. Sibirsk. Maz. Ž., 9, 1386–1394 (in Russian)
English translation (1968): Sib. Math. J., 9, 1039–1045
Seregin, G. (1985): Variational-difference scheme for problems in the mechanics of ideally elastoplastic media. Zh. Vychisl. Mat. Fiz., 25, 237–352 (in Russian)
English translation (1985): U.S.S.R Comp. Math. and Math. Phys., 25, 153–165
Seregin, G.(1990): Differential properties of solutions of variational problems for functionals with linear growth. Problemy Matematicheskogo Analiza, Vypusk 11, Isazadel’stvo LGU, 51–79 (in Russian)
English translation (1993): J. Soviet Math., 64, 1256–1277
Seregin, G. (1996): Twodimensional variational problems in plasticity theory. Izv. Russian Academy of Sciences, 60, 175–210 (in Russian) English translation (1996): Izvestiya Mathematics, 60, 179-216
Strang, G., Temam, R. (1980): Duality and relaxations in the theory of plasticity. J. Méchanique,19, 1–35
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bildhauer, M., Fuchs, M. (2003). Convex Variational Problems with Linear Growth. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-55627-2_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44051-2
Online ISBN: 978-3-642-55627-2
eBook Packages: Springer Book Archive