On the interplay between coefficients in some nonlinear dirichlet problems with distributional data

Prof. David Arcoya

Date: 19th February 10:00 am - 10:45 am (UTC -3) Link: PPGM/UFSCar Youtube channel

Abstract: We present the main results of the joint paper [1] with Lucio Boccardo and Luigi Orsina. Specifically, we prove the exponential summability of the solution u of the Dirichlet problem

u∈W1,2(Ω):−div(M(x,∇u))+a(x)u=−div(F). 0

where Ω is a bounded set in RN and −div(M (x,∇u)) is a classical nonlinear differential operator, defined by a Carathéodory function M(x,ξ) satisfying, for some 0 < α ≤ β, and for almost every x ∈ Ω,

M(x,ξ)ξ≥α|ξ|2 , |M(x,ξ)|≤β|ξ|, ∀ξ ∈ RN ,

[M(x,ξ)−M(x,η)](ξ−η)>0, ∀ξ,η∈RN , ξ̸=η.

Our key assumption is that the function 0 ≤a(x)∈L1(Ω) and the vector-valued function F(x) are such that

∃R > 0 such that |F(x)|2≤Ra(x).

In addition, we prove the boundedness of u under the slightly stronger assumption that

∃R>0 and ∃p>2 such that |F(x)|p≤Ra(x).


[1] Arcoya, D., Boccardo, L., Orsina, L. Regularizing effect of the interplay between coefficients in some nonlinear Dirichlet problems with distributional data. Annali di Matematica 199, 1909–1921 (2020).