Integral inequalities and summability of solutions of some differential problems

• Autorzy: Lucio Boccardo
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2000
• Tom: 52
• Numer: 1
• Strony: 25-28

Daty

wydano

2000

Twórcy

autor

Lucio Boccardo

• Dipartimento di Matematica, Università di Roma I, Piazza A. Moro 2, 00185, Roma, Italy

Bibliografia

• [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).
• [2] L. Boccardo and D. Giachetti, Existence results via regularity for some nonlinear elliptic problems, Comm. Partial Differential Equations 14 (1989), 663-680.
• [3] L. Boccardo and D. Giachetti, $L^s$-regularity of solutions of some nonlinear elliptic problems, preprint.
• [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.
• [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.
• [6] D. Giachetti and M. M. Porzio, Local regularity results for minima of functionals of Calculus of Variations, Nonlinear Anal., to appear.
• [7] M. M. Porzio, Local regularity results for some parabolic equations, preprint.
• [8] L. Boccardo and T. Gallouet, Nonlinear elliptic equations with right hand side measures, Comm. P.D.E. 17 (1992), 641-655.
• [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.
• [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.