Existence and uniqueness results for solutions of nonlinear equations with right hand side in L1
Fiorenza, A. ; Sbordone, C.
Studia Mathematica, Tome 129 (1998), p. 223-231 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here f∈L1(Ω) and the solution belongs to the so-called grand Sobolev space W01,2)(Ω). This is the proper space when the right hand side is assumed to be only L1-integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:216469 
@article{bwmeta1.element.bwnjournal-article-smv127i3p223bwm,
     author = {A. Fiorenza and C. Sbordone},
     title = {Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$
            },
     journal = {Studia Mathematica},
     volume = {129},
     year = {1998},
     pages = {223-231},
     zbl = {0891.35039},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv127i3p223bwm}
}

Fiorenza, A.; Sbordone, C. Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$
            . Studia Mathematica, Tome 129 (1998) pp. 223-231. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv127i3p223bwm/

[00000] [B] L. Boccardo, manuscript, 1995.

[00001] [BB] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa 22 (1995), 241-273. | Zbl 0866.35037

[00002] [BG] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. | Zbl 0707.35060

[00003] [BM] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of -Δu=V(x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253. | Zbl 0746.35006

[00004] [CL] S. Chanillo and Y. Y. Li, Continuity of solutions of uniformly elliptic equations in ℝ2, Manuscripta Math. 77 (1992), 415-433. | Zbl 0797.35031

[00005] [CS] M. Carozza and C. Sbordone, The distance to L∞ in some function spaces and applications, Differential Integral Equations 10 (1997), 599-607. | Zbl 0889.35027

[00006] [D] T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185-203. | Zbl 0815.35023

[00007] [FLS] N. Fusco, P. L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), 561-565. | Zbl 0841.46023

[00008] [G] L. Greco, A remark on the equality det Df = Det Df, Differential Integral Equations 6 (1993), 1089-1100. | Zbl 0784.49013

[00009] [GIS] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math. 92 (1997), 249-258. | Zbl 0869.35037

[00010] [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. | Zbl 0562.35001

[00011] [IS1] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129-143. | Zbl 0766.46016

[00012] [IS2] T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew Math. 454 (1994), 143-161. | Zbl 0802.35016

[00013] [LL] J. Leray et J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97-107. | Zbl 0132.10502

[00014] [LM] P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, to appear.

[00015] [M] F. Murat, Conference at Pont à Mousson, 1994.

[00016] [Z] W. D. Ziemer, Weakly Differentiable Functions, Springer, 1989. | Zbl 0692.46022