A semilinear elliptic problem with a singularity in \(u = 0\)

Fran├žois Murat

(UPCM, Paris, France)

Event: ERC Workshop on Existence and Regularity for Nonlinear Systems of Partial Differential Equations

Date: Jul 04, 2014, time: 10:50

Abstract. In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem
\[ - div A(x) Du = F (x, u) \; {\rm in} \; \Omega,\]
\[ u = 0 \; {\rm on} \; \partial \Omega,\]
(namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition),
where the non\-linearity \(F (x, u)\) is singular in \(u = 0\), with a singularity of the type
\[F (x, u) = {f(x) \over u^\gamma} + g(x)\]
with \(\gamma > 0\) and \(f\) and \(g\) non negative (which implies that also \(u\) is non negative).

The main difficulty is to give a convenient definition of the solution of the problem, in particular when \(\gamma > 1\). We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when \(F(x, u)\) is non increasing en \(u\).

We also consider the homogenization problem where \(\Omega\) is replaced by \(\Omega^\varepsilon\), with \(\Omega^\varepsilon\) obtained from \(\Omega\) by removing many very
small holes in such a way that passing to the limit when \(\varepsilon\) tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" \(\mu u\) appears.

This work was inspired by the paper of Lucio Boccardo and Luigi Orsina \( Semilinear\) \(elliptic\) \(equations\) \(with\) \(singular\) \(nonlinearities\), Calc. Var. Partial Differential Equations, 37, (2010), pp. 363--380.