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Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms


Díaz Díaz, Jesús Ildefonso and Boccardo, L. and Giachetti, D. and Murat, F. (1993) Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. Journal of Differential Equations, 106 (2). pp. 215-237. ISSN 0022-0396

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The authors study the nonlinear elliptic equation
(*) −div(a(x,u,Du))−div(Φ(u))+g(x,u)=f(x)in Ω
with the boundary condition (∗∗) u=0 on ∂Ω, where Ω is a bounded open subset of RN, A(u)=−div(a(x,u,Du)) is a nonlinear operator of Leray-Lions type from W1,p0(Ω) into its dual, Φ∈(C0(R))N, g(x,t)t≥0, and f∈W−1,p′(Ω). No growth hypothesis is assumed on the vector-valued function Φ(u). The term div(Φ(u)) may be meaningless for usual weak solutions, even as a distribution. The authors deal with a weaker form of (∗),(∗∗), solutions of which in W1,p0(Ω) are called the "renormalized solutions'' of the original problem (∗),(∗∗). This weaker form can be formally obtained from (∗) by means of pointwise multiplication by h(u), where h is a C1 function with compact support. They prove the existence of renormalized solutions of (∗),(∗∗) and give sufficient conditions under which a renormalized solution is a usual weak solution. Furthermore, the authors study Ls- and L∞-regularity of renormalized solutions.

Item Type:Article
Uncontrolled Keywords:renormalized solutions; nonlinear Leray-Lions operator; largest class of possible test functions
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
Sciences > Mathematics > Numerical analysis
ID Code:16166

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