Gradient estimates and Harnack inequalities for solutions to the minimal surface equation

Mario Miranda

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: 1, page 27-30
  • ISSN: 1120-6330

Abstract

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A gradient estimate for solutions to the minimal surface equation can be proved by Partial Differential Equations methods, as in [2]. In such a case, the oscillation of the solution controls its gradient. In the article presented here, the estimate is derived from the Harnack type inequality established in [1]. In our case, the gradient is controlled by the area of the graph of the solution or by the integral of it. These new results are similar to the one announced by Ennio De Giorgi in [3].

How to cite

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Miranda, Mario. "Gradient estimates and Harnack inequalities for solutions to the minimal surface equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.1 (2000): 27-30. <http://eudml.org/doc/252385>.

@article{Miranda2000,
abstract = {A gradient estimate for solutions to the minimal surface equation can be proved by Partial Differential Equations methods, as in [2]. In such a case, the oscillation of the solution controls its gradient. In the article presented here, the estimate is derived from the Harnack type inequality established in [1]. In our case, the gradient is controlled by the area of the graph of the solution or by the integral of it. These new results are similar to the one announced by Ennio De Giorgi in [3].},
author = {Miranda, Mario},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Minimal surface; Gradient estimate; Harnack inequality; Generalized solution for the minimal surface equation; minimal surface; gradient estimate},
language = {eng},
month = {3},
number = {1},
pages = {27-30},
publisher = {Accademia Nazionale dei Lincei},
title = {Gradient estimates and Harnack inequalities for solutions to the minimal surface equation},
url = {http://eudml.org/doc/252385},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Miranda, Mario
TI - Gradient estimates and Harnack inequalities for solutions to the minimal surface equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/3//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 1
SP - 27
EP - 30
AB - A gradient estimate for solutions to the minimal surface equation can be proved by Partial Differential Equations methods, as in [2]. In such a case, the oscillation of the solution controls its gradient. In the article presented here, the estimate is derived from the Harnack type inequality established in [1]. In our case, the gradient is controlled by the area of the graph of the solution or by the integral of it. These new results are similar to the one announced by Ennio De Giorgi in [3].
LA - eng
KW - Minimal surface; Gradient estimate; Harnack inequality; Generalized solution for the minimal surface equation; minimal surface; gradient estimate
UR - http://eudml.org/doc/252385
ER -

References

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  1. Bombieri, E. - Giusti, E., Harnack’s inequality for elliptic differential equations on minimal surfaces. Invent. Math., 15, 1972, 24-46. Zbl0227.35021MR308945
  2. Bombieri, E. - De Giorgi, E. - Miranda, M., Una maggiorazione a priori relativa alle ipersuperficie minimali non parametriche. Arch. Rat. Mech. Anal., 32, 1969, 255-267. Zbl0184.32803MR248647
  3. De Giorgi, E., Maggiorazioni a priori relative ad ipersuperfici minimali. Ist. Naz. di Alta Matematica, Symposia Mathematica II, 1968, 282-284. 
  4. Massari, U. - Miranda, M., Minimal surfaces of codimension one. Math. Studies, 91, North Holland Publ. Co., 1984. Zbl0565.49030MR795963
  5. Miranda, M., Superfici minime illimitate. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. IV, 4, 1977, 313-322. Zbl0352.49020MR500423
  6. Miranda, M., Maximum principles and minimal surfaces. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. IV, 25, 1997, 667-681. Zbl1015.49028MR1655536

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