Pointwise Gradient Estimates of Glaeser's Type

Italo Capuzzo Dolcetta; Antonio Vitolo

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 2, page 211-224
  • ISSN: 0392-4041

Abstract

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In this paper we are concerned with gradient estimates for viscosity solutions of fully nonlinear second order elliptic equations, generalizing to the nonlinear setting the results of Yanyan Li and Louis Nirenberg about the so-called Glaeser estimate and improving the qualitative results contained in one of our preceding papers.

How to cite

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Capuzzo Dolcetta, Italo, and Vitolo, Antonio. "Pointwise Gradient Estimates of Glaeser's Type." Bollettino dell'Unione Matematica Italiana 5.2 (2012): 211-224. <http://eudml.org/doc/290943>.

@article{CapuzzoDolcetta2012,
abstract = {In this paper we are concerned with gradient estimates for viscosity solutions of fully nonlinear second order elliptic equations, generalizing to the nonlinear setting the results of Yanyan Li and Louis Nirenberg about the so-called Glaeser estimate and improving the qualitative results contained in one of our preceding papers.},
author = {Capuzzo Dolcetta, Italo, Vitolo, Antonio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {211-224},
publisher = {Unione Matematica Italiana},
title = {Pointwise Gradient Estimates of Glaeser's Type},
url = {http://eudml.org/doc/290943},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Capuzzo Dolcetta, Italo
AU - Vitolo, Antonio
TI - Pointwise Gradient Estimates of Glaeser's Type
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/6//
PB - Unione Matematica Italiana
VL - 5
IS - 2
SP - 211
EP - 224
AB - In this paper we are concerned with gradient estimates for viscosity solutions of fully nonlinear second order elliptic equations, generalizing to the nonlinear setting the results of Yanyan Li and Louis Nirenberg about the so-called Glaeser estimate and improving the qualitative results contained in one of our preceding papers.
LA - eng
UR - http://eudml.org/doc/290943
ER -

References

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  2. CAFFARELLI, L. A. - CABRÈ, X. , Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43 (1995), Providence, Rhode Island. MR1351007DOI10.1090/coll/043
  3. CAPUZZO DOLCETTA, I. - VITOLO, A., Gradient and Hölder estimates for positive solutions of Pucci type equations, C. R., Math., Acad. Sci. Paris, 346 (2008), 527-532. Zbl1145.35050MR2412790DOI10.1016/j.crma.2008.03.004
  4. CAPUZZO DOLCETTA, I. - VITOLO, A., C 1 , α and Glaeser type estimates, Rend. Mat., Ser VII, 29 (2009), 17-27. MR2548484
  5. CAPUZZO DOLCETTA, I. - VITOLO, A., Glaeser's type gradient estimates for non- negative solutions of fully nonlinear elliptic equations, Discrete ans Continuous Dynamical Systems-Series A (DCDS-A). A special issue Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday Part I, 28, n. 2 (October 2010), 539-557. Zbl1193.35043MR2644755DOI10.3934/dcds.2010.28.539
  6. CRANDALL, M. G. - ISHII, H. - LIONS, P. L., User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. Zbl0755.35015MR1118699DOI10.1090/S0273-0979-1992-00266-5
  7. GILBARG, D. - TRUDINGER, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften No. 224, Springer-Verlag, Berlin-New York (1983). Zbl0562.35001MR737190DOI10.1007/978-3-642-61798-0
  8. GLAESER, G., Racine carrée d'une fonction différentiable, Ann. Ist. Fourier, 13 (1963), 203-207. Zbl0128.27903MR163995
  9. HADAMARD, J., Sur certaines propriétés des trajectoires en dynamique, J. Math. Sér. 5, 3 (1897), 331-387. Zbl28.0643.01
  10. KOLMOGOROV, A. N., Une géneralization de l'inégalité de M. J. Hadamard entre les bornes supériores des dérivées successives d'une fonction, C. R. Acad. Sci. Paris, 207 (1963), 764-765. 
  11. LANDAU, E., Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1913), 43-49. Zbl44.0463.02MR1577513DOI10.1112/plms/s2-13.1.43
  12. LI, Y. Y. - NIRENBERG, L., Generalization of a well-known inequality, Progress in Nonlinear Differential Equations and Their Applications, 66 (2005), 365-370. Zbl1284.26021MR2187814DOI10.1007/3-7643-7401-2_24
  13. MAZ'YA, V. G. - KUFNER, A., Variations on the theme of the inequality ( f ′ ) 2 ≤ 2 ⁢ f ⁢ sup ⁡ | f ′′ | , Manuscripta Math., 56 (1986), 89-104. Zbl0605.26010MR846988DOI10.1007/BF01171035
  14. MAZ'YA, V. G. - SHAPOSHNIKOVA, T. O., Sharp pointwise interpolation inequalities for derivatives, Funct. Anal. Appl., 36 (2002), 30-48. Zbl1034.42018

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