elliptic partial differential equations

Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data

We prove boundedness and continuity for solutions to the Dirichlet problem for the equation
[
-Div(a(x,
abla u))=h(x,u)+mu,,quadhbox{in }OmegasubsetR^N,,
]
where the left--hand side is a Leray--Lions operator from $W_0^{1,p}$ into $W^{-1,p'}(Omega)$, with $1

Derivation of macroscopic equilibrium models for heat conduction in finely mixed composite media with singular sources

We prove existence and homogenization results for a family (depending on a small parameter and on a parameter 2 f1; 0; 1g) of elliptic problems involving a singular lower order term and representing the Euler equations of energy functionals, which can be used to describe the equilibrium for the heat conduction in composite materials with two finely mixed phases having a microscopic periodic structure (for details on the related physical models see for instance [3, 4] and the reference quoted there).