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oai:docta.ucm.es:20.500.14352/574842023-08-25T10:50:34Zcom_20.500.14352_14col_20.500.14352_15

Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms Díaz Díaz, Jesús Ildefonso Boccardo, L. Giachetti, D. Murat, F. 517.98 519.6 renormalized solutions nonlinear Leray-Lions operator largest class of possible test functions Análisis funcional y teoría de operadores Análisis numérico 1206 Análisis Numérico 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. Depto. de Análisis Matemático y Matemática Aplicada Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T16:56:46Z 2023-06-20T16:56:46Z 1993-11 journal article https://hdl.handle.net/20.500.14352/57484 0022-0396 10.1006/jdeq.1993.1106 eng restricted access application/pdf Elsevier