Hardy-Sobolev Inequalities for Hessian Integrals

Nunzia Gavitone

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 951-967
  • ISSN: 0392-4033

Abstract

top
Using appropriate symmetrization arguments, we prove the Hardy-Sobolev type inequalities for Hessian Integrals which extend the classical results, well known for Sobolev functions. For such inequalities the value of the best constant is given. Finally we give an improvement of these inequalities by adding a second term that, involves another singular weight which is a suitable negative power of log ( | x | ) .

How to cite

top

Gavitone, Nunzia. "Hardy-Sobolev Inequalities for Hessian Integrals." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 951-967. <http://eudml.org/doc/290419>.

@article{Gavitone2007,
abstract = {Using appropriate symmetrization arguments, we prove the Hardy-Sobolev type inequalities for Hessian Integrals which extend the classical results, well known for Sobolev functions. For such inequalities the value of the best constant is given. Finally we give an improvement of these inequalities by adding a second term that, involves another singular weight which is a suitable negative power of $\log (|x|)$.},
author = {Gavitone, Nunzia},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {951-967},
publisher = {Unione Matematica Italiana},
title = {Hardy-Sobolev Inequalities for Hessian Integrals},
url = {http://eudml.org/doc/290419},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Gavitone, Nunzia
TI - Hardy-Sobolev Inequalities for Hessian Integrals
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 951
EP - 967
AB - Using appropriate symmetrization arguments, we prove the Hardy-Sobolev type inequalities for Hessian Integrals which extend the classical results, well known for Sobolev functions. For such inequalities the value of the best constant is given. Finally we give an improvement of these inequalities by adding a second term that, involves another singular weight which is a suitable negative power of $\log (|x|)$.
LA - eng
UR - http://eudml.org/doc/290419
ER -

References

top
  1. BANDLE, C., Isoperimetric Inequalities and ApplicationsPitman, 1980. Zbl0436.35063MR572958
  2. BENNETT, C. - SHARPLEY, R., Interpolation of Operators, Academic Press, 1988. Zbl0647.46057MR928802
  3. BLISS, G.A., An integral inequality, J. London Math. Soc., 5 (1930), 40-46. Zbl56.0434.02MR1574997DOI10.1112/jlms/s1-5.1.40
  4. BREZIS, H. - VÁZQUEZ, J. L., Blow-up solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469. MR1605678
  5. BURAGO, Yu. D. - ZALGALLER, V. A., Geometric Inequalities, Springer-Verlag, 1988. MR936419DOI10.1007/978-3-662-07441-1
  6. CHAUDHURI, N. - ADIMURTHI, - RAMASWAMY, M., An improved Hardy-Sobolev inequality and its applications, Proc. Am. Math. Soc., 130, (2001), 489-505. Zbl0987.35049MR1862130DOI10.1090/S0002-9939-01-06132-9
  7. CHOU, K. S. - GENG, D., Critical dimension of a Hessian Equation Involving Critical Exponent and a Related Asymptotic Result, J. Differential Equations, 129 (1996), 79-110. Zbl0864.35037MR1400797DOI10.1006/jdeq.1996.0112
  8. EGNELL, H., Elliptic Boundary Value Problems with Singular Coefficients and Critical Nonlinearities, Indiana Univ. Math. J., 38 (1989), 235-251. Zbl0666.35072MR997382DOI10.1512/iumj.1989.38.38012
  9. GARCIA AZORERO, J. - PERAL ALONSO, I., Hardy inequalities and some critical elliptic parabolic problems, J. Diff. Eq., 144 (1998), 441-476. Zbl0918.35052MR1616905DOI10.1006/jdeq.1997.3375
  10. GILBARG, D. - TRUDINGER, N. S., Elliptic partial differential equation of second order, (Second Edition), Springer-Verlag, 1983. Zbl0562.35001MR737190DOI10.1007/978-3-642-61798-0
  11. MAZ'JA, V. G. - Sobolev Spaces, Springer-Verlag, 1985. MR817985DOI10.1007/978-3-662-09922-3
  12. OPIC, B. - KUFNER, A., Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Group UK Limited, London, 1990. Zbl0698.26007MR1069756
  13. REILLY, R. C., On the Hessian of a function and the curvatures of its graph, Michigan Math. J., 20 (1974), 373-383. MR334045
  14. TALENTI, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. Zbl0353.46018MR463908DOI10.1007/BF02418013
  15. TRUDINGER, N. S., On new isoperimetric inequalities and symmetrization, J. Reine Angew. Math., 488 (1997), 203-220. Zbl0883.52006MR1465371DOI10.1515/crll.1997.488.203
  16. TRUDINGER, N. S., Isoperimetric inequalities for quermassintegrals, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 11 (1994), 411-425. Zbl0859.52001MR1287239DOI10.1016/S0294-1449(16)30181-0
  17. TSO, K., Remarks on critical exponents for Hessian operators, Ann. Inst. H. Poincaré7 (1990), 113-122. Zbl0715.35031MR1051232DOI10.1016/S0294-1449(16)30302-X
  18. TSO, K., On symmetrization and Hessian Equations, J. Anal. Math., 52 (1989), 94-106. Zbl0675.35040MR981497DOI10.1007/BF02820473
  19. VÀZQUEZ, J. L. - ZUAZUA, E., The Hardy Inequality and the Asymptotic Behaviour of the Heat Equation with an Inverse-Square Potential, J. Func. Anal., 173 (2000), 103-153. Zbl0953.35053MR1760280DOI10.1006/jfan.1999.3556
  20. WANG, X. J., A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43 (1994), 25-54. Zbl0805.35036MR1275451DOI10.1512/iumj.1994.43.43002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.