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


Díaz Díaz, Jesús Ildefonso y Boccardo, L. y Giachetti, D. y 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.

Tipo de documento:Artículo
Palabras clave:renormalized solutions; nonlinear Leray-Lions operator; largest class of possible test functions
Materias:Ciencias > Matemáticas > Análisis funcional y teoría de operadores
Ciencias > Matemáticas > Análisis numérico
Código ID:16166

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