A new proof of the stability of renormalized solutions to elliptic equations with measure data

Abstract

One of the recent advances in the investigation on nonlinear elliptic equations with a measure as forcing term is a paper by G. Dal Maso, F. Murat, L. Orsina and A. Prignet in which it has been introduced the notion of renormalized solution to the problem $\[-\mathrm{div}(a(x,\nabla u))=\mu\quad\mbox{in }\varOmega,\qquad u=0\quad\mbox{on }\curpartial\varOmega.\]$ Here Ω is a bounded open set of R^N, N≥2, the operator is modelled on the p‐Laplacian, and μ is a Radon measure with bounded variation in Ω. The existence of a renormalized solution is obtained by approximation as a consequence of a stability result.

We provide a new proof of this stability result, based on the properties of the truncations of the renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.