Boundaries of prescribed mean curvature

Eduardo H. A. Gonzales; Umberto Massari; Italo Tamanini

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993)

  • Volume: 4, Issue: 3, page 197-206
  • ISSN: 1120-6330

Abstract

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The existence of a singular curve in R 2 is proven, whose curvature can be extended to an L 2 function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.

How to cite

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Gonzales, Eduardo H. A., Massari, Umberto, and Tamanini, Italo. "Boundaries of prescribed mean curvature." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.3 (1993): 197-206. <http://eudml.org/doc/244125>.

@article{Gonzales1993,
abstract = {The existence of a singular curve in \( \mathbb\{R\}^\{2\} \) is proven, whose curvature can be extended to an \( L^\{2\} \) function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.},
author = {Gonzales, Eduardo H. A., Massari, Umberto, Tamanini, Italo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Calculus of variations; Geometric measure theory; Mean curvature; Singular boundaries of finite measure; singular boundaries of finite measure; mean curvature},
language = {eng},
month = {9},
number = {3},
pages = {197-206},
publisher = {Accademia Nazionale dei Lincei},
title = {Boundaries of prescribed mean curvature},
url = {http://eudml.org/doc/244125},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Gonzales, Eduardo H. A.
AU - Massari, Umberto
AU - Tamanini, Italo
TI - Boundaries of prescribed mean curvature
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/9//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 3
SP - 197
EP - 206
AB - The existence of a singular curve in \( \mathbb{R}^{2} \) is proven, whose curvature can be extended to an \( L^{2} \) function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.
LA - eng
KW - Calculus of variations; Geometric measure theory; Mean curvature; Singular boundaries of finite measure; singular boundaries of finite measure; mean curvature
UR - http://eudml.org/doc/244125
ER -

References

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  1. BAROZZI, E., The curvature of a boundary with finite area. Preprint Dip. di Mat., Università di Trento, 1993. 
  2. BAROZZI, E. - GONZALEZ, E. H. A. - TAMANINI, I., The mean curvature of a set of finite perimeter. Proc. A.M.S., 99, 1987, 313-316. MR870791DOI10.2307/2046631
  3. BAROZZI, E. - TAMANINI, I., Penalty methods for minimal surfaces with obstacles. Ann. Mat. Pura Appl., 152, (IV), 1988, 139-157. Zbl0824.49035MR980976DOI10.1007/BF01766145
  4. CONGEDO, G. - TAMANINI, I., Note sulla regolarità dei minimi di funzionali del tipo dell'area. Rend. Accad. Naz. XL, vol. XII, fasc. 17, 106, 1988, 239-257. Zbl0674.49031MR985069
  5. GIUSTI, E., Minimal surfaces and functions of bounded variation. Birkhäuser, Boston-Basel-Stuttgart1984. Zbl0545.49018MR775682
  6. MASSARI, U., Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in R n . Arch. Rat. Mech. An., 55, 1974, 357-382. Zbl0305.49047MR355766
  7. MASSARI, U., Frontiere orientate di curvatura media assegnata in L p . Rend. Sem. Mat. Univ. Padova, 53, 1975, 37-52. Zbl0358.49019MR417905
  8. MASSARI, U. - MIRANDA, M., Minimal Surface of Codimension One. North Holland, Amsterdam1984. Zbl0565.49030MR795963
  9. MASSARI, U. - PEPE, L., Successioni convergenti di ipersuperfici di curvatura media assegnata. Rend. Sem. Mat. Univ. Padova, 53, 1975, 53-68. Zbl0358.49020MR420401
  10. TAMANINI, I., Regularity results for almost minimal oriented hypersurfaces in R n . Quaderni del Dipartimento di Matematica, Univ. Lecce, n° 1, 1984. Zbl1191.35007
  11. TAMANINI, I., Interfaces of prescribed mean curvature. In: P. CONCUS, R. FINN (eds.), Variational Methods for Free Surface Interface. Springer-Verlag, New York-Berlin-Heidelberg1987, 91-97. MR872892

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