English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We perform the homogenization of the semilinear elliptic problem  \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \;\Omega^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon =F(x,u^\varepsilon) & \mbox{in} \; \Omega^\varepsilon,\\ u^\varepsilon = 0 &\mbox{on} \; \partial \Omega^\varepsilon.\\ \end{cases} \end{equation*} In thisproblem $F(x,s)$ is a Carath\'eodory function such that $0 \leq F(x,s) \leqh(x)/\Gamma(s)$ a.e. $x\in\Omega$ for every $s > 0$, with $h$ in some$L^r(\Omega)$ and $\Gamma$ a $C^1([0, +\infty[)$ function such that $\Gamma(0)= 0$ and $\Gamma'(s) > 0$ for every $s > 0$. On the other hand the open sets$\Omega^\varepsilon$ are obtained by removing many small holes from a fixedopen set $\Omega$ in such a way that a "strange term" $\mu u^0$ appears in thelimit equation in the case where the function $F(x,s)$ depends only on $x$.  We already treated this problem in the case of a "mild singularity", namelyin the case where the function $F(x,s)$ satisfies $0 \leq F(x,s) \leq h(x)(\frac 1s + 1)$. In this case the solution $u^\varepsilon$ to the problembelongs to $H^1_0 (\Omega^\varepsilon)$ and its definition is a "natural" andrather usual one.  In the general case where $F(x,s)$ exhibits a "strong singularity" at $u =0$, which is the purpose of the present paper, the solution $u^\varepsilon$ tothe problem only belongs to $H_{\tiny loc}^1(\Omega^\varepsilon)$ but ingeneral does not belongs to $H^1_0 (\Omega^\varepsilon)$ any more, even if$u^\varepsilon$ vanishes on $\partial\Omega^\varepsilon$ in some sense.Therefore we introduced a new notion of solution (in the spirit of thesolutions defined by transposition) for problems with a strong singularity.This definition allowed us to obtain existence, stability and uniquenessresults.

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at $u=0$ in a domain with many small holes