### Impacto

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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Official URL: http://www.sciencedirect.com/science/article/pii/S002203968371106X

## Abstract

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 |
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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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Deposited On: | 11 Sep 2012 08:16 |

Last Modified: | 06 Feb 2014 10:39 |

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