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

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

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topMiranda, 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

top- Bombieri, E. - Giusti, E., Harnack’s inequality for elliptic differential equations on minimal surfaces. Invent. Math., 15, 1972, 24-46. Zbl0227.35021MR308945
- 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
- De Giorgi, E., Maggiorazioni a priori relative ad ipersuperfici minimali. Ist. Naz. di Alta Matematica, Symposia Mathematica II, 1968, 282-284.
- Massari, U. - Miranda, M., Minimal surfaces of codimension one. Math. Studies, 91, North Holland Publ. Co., 1984. Zbl0565.49030MR795963
- Miranda, M., Superfici minime illimitate. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. IV, 4, 1977, 313-322. Zbl0352.49020MR500423
- 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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