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Potential Analysis

, Volume 24, Issue 1, pp 15–45 | Cite as

A Fully Nonlinear Nonhomogeneous Neumann Problem

Dedicated to the memory of the Professor Philippe Bénilan
  • Abdelmajid Siai
Article

Abstract

Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, fL 1(ℝN), gL 1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
$$\mathbf{(P)}\quad \left\{\begin{array}{l}-\mathrm{div}[a(x,\nabla u)]+\beta(u)=f\quad \mbox{in}\ \mathbb{R}^{N}\setminus \partial \Omega,\cr \bigl[\frac{\partial u}{\partial \nu_{a}}\big]+\gamma(\tau u)=g\quad \hbox{on}\ \partial \Omega,\cr [u]=0\end{array}\right.$$
in the sense that, if T k(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DT k u on the set {|u|<k}) and \(\mathcal{T}^{1,p}(\mathbb{R}^{N})=\{u:\mathbb{R}^{N}\rightarrow \mathbb{R}\) , u measurable; DT k(u)∈L p(ℝN), k>0}, then \(u\in \mathcal{T}^{1,p}(\mathbb{R}^{N})\) and u satisfies,
$$\int_{\mathbb{R}^{N}}\langle a(\cdot,Du),DT_{k}(u-\varphi)\rangle \leqslant \int_{\mathbb{R}^{N}}(f-\beta(u))T_{k}(u-\varphi)+\int_{\partial \Omega}(g-\gamma(u))T_{k}(\tau u-\tau\varphi),$$
for every k>0 and every \(\varphi\in \mathcal{T}^{1,p}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) .

Keywords

nonlinear Neumann problem elliptic operator of Leray Lions type m-completely accretive operator 

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References

  1. 1.
    Andreu, F., Mazón, J.M., Sigura de León, S. and Toledo, J.: ‘Quasi-linear elliptic and parabolic equations in L 1 with non-linear boundary conditions’, Adv. Math. Sci. Appl. 7 (1997), 183–213. MathSciNetMATHGoogle Scholar
  2. 2.
    Bénilan, P.: ‘Equations d’évolution dans un espace de Banach quelconque et applications', Thèse, Univ. Orsay, 1972. Google Scholar
  3. 3.
    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J.L.: ‘An L 1- theory of existence and uniqueness of solutions of nonlinear elliptic equations’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2) (1995), 241–273. MathSciNetMATHGoogle Scholar
  4. 4.
    Bénilan, P., Brézis, H. and Crandall, M.G.: ‘A semilinear equation in L 1’, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. 2 (1975), 523–555. MathSciNetMATHGoogle Scholar
  5. 5.
    Bénilan, Ph. and Crandall, M.G.: ‘Completely accretive operators’, in Ph. Clement et al. (eds), Semigroup Theory and Evolution Equations, Marcel Dekker, New York, 1991, pp. 41–76. Google Scholar
  6. 6.
    Boccardo, L. and Gallouët, T.: ‘Nonlinear elliptic equations with right-hand side measures’, Comm. Partial Differential Equations 17 (1992), 641–655. MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brézis, H.: ‘Equations et inequations non linéaires dans les espaces vectoriels en dualité’, Ann. Inst. Fourier 18 (1968), 115–175. MATHGoogle Scholar
  8. 8.
    Brézis, H. and Strauss, W.: ‘Semi-linear second order elliptic equation in L 1’, J. Math. Soc. Japan 25 (1973), 565–590. MathSciNetMATHGoogle Scholar
  9. 9.
    Casas, E. and Fernandez, L.A.: ‘A Green formula for quasilinear elliptic operators’, J. Math. Anal. Appl. 142 (1989), 62–73. CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Cwikel, M. and Fefferman, C.: ‘The canonical semi-norm in weak L 1’, Studia Math. 78 (1984). Google Scholar
  11. 11.
    Heinonen, J., Kilpeläinen, T. and Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993. MATHGoogle Scholar
  12. 12.
    Iwaniec, T. and Sbordone, C.: ‘Weak minima of variational integrals’, J. Reine Angew. Math. 454 (1994), 143–161. MathSciNetMATHGoogle Scholar
  13. 13.
    Leray, J. and Lions, J.-L.: ‘Quelques résultats de Vi \(\) ik sur les problèmes elliptiques non linéaires par les méthodes de Minty–Browder’, Bull. Soc. Math. France 93 (1965), 97–107. Google Scholar
  14. 14.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gautier-Villars, Paris, 1969. MATHGoogle Scholar
  15. 15.
    Murat, F.: Proceeding of the International Conferences on Nonlinear Analysis, Besançon, June 1994. Google Scholar
  16. 16.
    Nečas, J.: Les méthodes directes en théorie des equations elliptiques, Masson et Cie, Paris, 1967. Google Scholar
  17. 17.
    Prignet, A.: ‘Problèmes elliptiques et paraboliques dans un cadre non variationnel’, Thèse, 1996. Google Scholar
  18. 18.
    Serrin, J.: ‘Pathological solutions of elliptic differential equations’, Ann. Scuola Norm. Pisa. Cl. Sci. (1964), 385–387. Google Scholar
  19. 19.
    Siai, A.: ‘Régularité des traces sur ℝN−1×{0} des solutions du problème −Δu=f+g⊗δ (fL 1(ℝN),gL 1(ℝN−1)) et applications’, Note présentée par Jacques-Louis Lions, C.R. Acad. Sci. Paris 297 (28 novembre 1983). Google Scholar
  20. 20.
    Siai, A.: ‘On a quasilinear elliptic partial differential equation of Thomas–Fermi type’, Boll. Un. Mat. Ital. (6) 4B (1985), 685–707. MathSciNetGoogle Scholar
  21. 21.
    Vazquez, J.L.: ‘Entropy solutions and the uniqueness problem for nonlinear second-order elliptic equations’, in A. Ben Kirane and J. P. Gossez (eds), Nonlinear Partial Differential Equations 343, Addison-Wesley Longman, New York, 1996, pp. 179–203. Google Scholar
  22. 22.
    Yosida, K.: Functional Analysis, 3rd edn, Springer-Verlag, New York, 1971. MATHGoogle Scholar
  23. 23.
    Zeidler, E.: Nonlinear Functional Analysis and Its applications, Vol. IV. Applications to Mathematical Physics, Springer, New York, 1997. Google Scholar
  24. 24.
    Ziemer, W.P.: Weakly Differential Functions, Grad. Texts in Math. 120, Springer-Verlag, 1989. Google Scholar

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© Springer 2006

Authors and Affiliations

  1. 1.Institut Préparatoire aux Études d'Ingénieurs de NabeulNabeulTunisia

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