Integral inequalities and summability of solutions of some differential problems

Lucio Boccardo

Banach Center Publications (2000)

  • Volume: 52, Issue: 1, page 25-28
  • ISSN: 0137-6934

Abstract

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The aim of this note is to indicate how inequalities concerning the integral of | ∇ u | 2 on the subsets where |u(x)| is greater than k ( k ∈ I R + ) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of | ∇ u | 2 on the subsets where |u(x)| is less than k ( k ∈ I R + ) or where k ≤ |u(x)| < k+1 were used in order to prove estimates in Sobolev spaces larger than W 0 1 , 2 ( Ω ) for solutions of Dirichlet problems with irregular data.

How to cite

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Boccardo, Lucio. "Integral inequalities and summability of solutions of some differential problems." Banach Center Publications 52.1 (2000): 25-28. <http://eudml.org/doc/209061>.

@article{Boccardo2000,
abstract = {The aim of this note is to indicate how inequalities concerning the integral of $|∇u|^2$ on the subsets where |u(x)| is greater than k ($k ∈ IR^+$) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of $|∇u|^2$ on the subsets where |u(x)| is less than k ($k ∈ IR^+$) or where k ≤ |u(x)| < k+1 were used in order to prove estimates in Sobolev spaces larger than $W^\{1,2\}_\{0\}(Ω)$ for solutions of Dirichlet problems with irregular data.},
author = {Boccardo, Lucio},
journal = {Banach Center Publications},
keywords = {Marcinkiewicz spaces; regularity; Dirichlet problems},
language = {eng},
number = {1},
pages = {25-28},
title = {Integral inequalities and summability of solutions of some differential problems},
url = {http://eudml.org/doc/209061},
volume = {52},
year = {2000},
}

TY - JOUR
AU - Boccardo, Lucio
TI - Integral inequalities and summability of solutions of some differential problems
JO - Banach Center Publications
PY - 2000
VL - 52
IS - 1
SP - 25
EP - 28
AB - The aim of this note is to indicate how inequalities concerning the integral of $|∇u|^2$ on the subsets where |u(x)| is greater than k ($k ∈ IR^+$) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of $|∇u|^2$ on the subsets where |u(x)| is less than k ($k ∈ IR^+$) or where k ≤ |u(x)| < k+1 were used in order to prove estimates in Sobolev spaces larger than $W^{1,2}_{0}(Ω)$ for solutions of Dirichlet problems with irregular data.
LA - eng
KW - Marcinkiewicz spaces; regularity; Dirichlet problems
UR - http://eudml.org/doc/209061
ER -

References

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  1. [1] L. Boccardo and D. Giachetti, Some remarks on the regularity of solutions of strongly nonlinear problems and applications, Ricerche Mat. 34 (1985), 309-323 (in Italian). Zbl0627.35034
  2. [2] L. Boccardo and D. Giachetti, Existence results via regularity for some nonlinear elliptic problems, Comm. Partial Differential Equations 14 (1989), 663-680. Zbl0678.35035
  3. [3] L. Boccardo and D. Giachetti, L s -regularity of solutions of some nonlinear elliptic problems, preprint. Zbl0627.35034
  4. [4] L. Boccardo, A. Dall'Aglio, T. Gallouet and L. Orsina, Existence and regularity results for some nonlinear parabolic equations, Adv. Math. Sci. Appl., to appear. Zbl0962.35093
  5. [5] L. Boccardo, E. Ferone, N. Fusco and L. Orsina, Regularity of minimizing sequences for functionals of the Calculus of Variations via the Ekeland principle, Differential Integral Eq. 12 (1999), 119-135. Zbl1007.49024
  6. [6] D. Giachetti and M. M. Porzio, Local regularity results for minima of functionals of Calculus of Variations, Nonlinear Anal., to appear. Zbl0942.49029
  7. [7] M. M. Porzio, Local regularity results for some parabolic equations, preprint. 
  8. [8] L. Boccardo and T. Gallouet, Nonlinear elliptic equations with right hand side measures, Comm. P.D.E. 17 (1992), 641-655. Zbl0812.35043
  9. [9] P. Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J. L. Vazquez, An L 1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali Sc. Norm. Sup. Pisa 22 (1995), 241-273. Zbl0866.35037
  10. [10] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189-258. Zbl0151.15401

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