# Existence and uniqueness results for solutions of nonlinear equations with right hand side in ${L}^{1}$

Studia Mathematica (1998)

- Volume: 127, Issue: 3, page 223-231
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topFiorenza, A., and Sbordone, C.. "Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$." Studia Mathematica 127.3 (1998): 223-231. <http://eudml.org/doc/216469>.

@article{Fiorenza1998,

abstract = {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 $f ∈ L^1(Ω)$ and the solution belongs to the so-called grand Sobolev space $W_0^\{1,2)\}(Ω)$. This is the proper space when the right hand side is assumed to be only $L^1$-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.},

author = {Fiorenza, A., Sbordone, C.},

journal = {Studia Mathematica},

keywords = {grand Sobolev space; exponential integrability},

language = {eng},

number = {3},

pages = {223-231},

title = {Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$},

url = {http://eudml.org/doc/216469},

volume = {127},

year = {1998},

}

TY - JOUR

AU - Fiorenza, A.

AU - Sbordone, C.

TI - Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$

JO - Studia Mathematica

PY - 1998

VL - 127

IS - 3

SP - 223

EP - 231

AB - 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 $f ∈ L^1(Ω)$ and the solution belongs to the so-called grand Sobolev space $W_0^{1,2)}(Ω)$. This is the proper space when the right hand side is assumed to be only $L^1$-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.

LA - eng

KW - grand Sobolev space; exponential integrability

UR - http://eudml.org/doc/216469

ER -

## References

top- [B] L. Boccardo, manuscript, 1995.
- [BB] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An ${L}^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa 22 (1995), 241-273. Zbl0866.35037
- [BG] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. Zbl0707.35060
- [BM] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u=V\left(x\right){e}^{u}$ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253. Zbl0746.35006
- [CL] S. Chanillo and Y. Y. Li, Continuity of solutions of uniformly elliptic equations in ${\mathbb{R}}^{2}$, Manuscripta Math. 77 (1992), 415-433. Zbl0797.35031
- [CS] M. Carozza and C. Sbordone, The distance to ${L}^{\infty}$ in some function spaces and applications, Differential Integral Equations 10 (1997), 599-607. Zbl0889.35027
- [D] T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185-203. Zbl0815.35023
- [FLS] N. Fusco, P. L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), 561-565. Zbl0841.46023
- [G] L. Greco, A remark on the equality det Df = Det Df, Differential Integral Equations 6 (1993), 1089-1100. Zbl0784.49013
- [GIS] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math. 92 (1997), 249-258. Zbl0869.35037
- [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. Zbl0562.35001
- [IS1] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129-143. Zbl0766.46016
- [IS2] T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew Math. 454 (1994), 143-161. Zbl0802.35016
- [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. Zbl0132.10502
- [LM] P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, to appear.
- [M] F. Murat, Conference at Pont à Mousson, 1994.
- [Z] W. D. Ziemer, Weakly Differentiable Functions, Springer, 1989. Zbl0692.46022

## Citations in EuDML Documents

top- Menita Carozza, Antonia Passarelli di Napoli, On very weak solutions of a class of nonlinear elliptic systems
- Menita Carozza, Gioconda Moscariello, Antonia Passarelli di Napoli, Linear elliptic equations with BMO coefficients
- M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in ${L}^{1}\left(\Omega \right)$
- M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in (Ω)
- Takao Ohno, Tetsu Shimomura, Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces
- Tadeusz Iwaniec, Carlo Sbordone, Quasiharmonic fields
- Gianni Dal Maso, François Murat, Luigi Orsina, Alain Prignet, Renormalized solutions of elliptic equations with general measure data
- Tadeusz Iwaniec, Carlo Sbordone, Caccioppoli estimates and very weak solutions of elliptic equations

## NotesEmbed ?

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