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1998 | 127 | 3 | 223-231
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
127
Numer
3
Strony
223-231
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-07-11
poprawiono
1997-07-04
Twórcy
autor
autor
Bibliografia
  • [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.
  • [BG] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169.
  • [BM] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-Δu = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253.
  • [CL] S. Chanillo and Y. Y. Li, Continuity of solutions of uniformly elliptic equations in $ℝ^2$, Manuscripta Math. 77 (1992), 415-433.
  • [CS] M. Carozza and C. Sbordone, The distance to $L^∞$ in some function spaces and applications, Differential Integral Equations 10 (1997), 599-607.
  • [D] T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185-203.
  • [FLS] N. Fusco, P. L. Lions and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), 561-565.
  • [G] L. Greco, A remark on the equality det Df = Det Df, Differential Integral Equations 6 (1993), 1089-1100.
  • [GIS] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math. 92 (1997), 249-258.
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
  • [IS1] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129-143.
  • [IS2] T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew Math. 454 (1994), 143-161.
  • [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.
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv127i3p223bwm
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