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

A. Fiorenza; C. Sbordone

Studia Mathematica (1998)

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

Abstract

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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.

How to cite

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Fiorenza, 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

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  4. [BM] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of - Δ u = V ( x ) e u in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253. Zbl0746.35006
  5. [CL] S. Chanillo and Y. Y. Li, Continuity of solutions of uniformly elliptic equations in 2 , Manuscripta Math. 77 (1992), 415-433. Zbl0797.35031
  6. [CS] M. Carozza and C. Sbordone, The distance to L in some function spaces and applications, Differential Integral Equations 10 (1997), 599-607. Zbl0889.35027
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  12. [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
  13. [IS2] T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew Math. 454 (1994), 143-161. Zbl0802.35016
  14. [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
  15. [LM] P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, to appear. 
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  17. [Z] W. D. Ziemer, Weakly Differentiable Functions, Springer, 1989. Zbl0692.46022

Citations in EuDML Documents

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  1. Menita Carozza, Antonia Passarelli di Napoli, On very weak solutions of a class of nonlinear elliptic systems
  2. Menita Carozza, Gioconda Moscariello, Antonia Passarelli di Napoli, Linear elliptic equations with BMO coefficients
  3. 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 ( Ω )
  4. Takao Ohno, Tetsu Shimomura, Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces
  5. 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)
  6. Tadeusz Iwaniec, Carlo Sbordone, Quasiharmonic fields
  7. Gianni Dal Maso, François Murat, Luigi Orsina, Alain Prignet, Renormalized solutions of elliptic equations with general measure data
  8. Tadeusz Iwaniec, Carlo Sbordone, Caccioppoli estimates and very weak solutions of elliptic equations

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