M. Dalla Riva | Searcharxiv

arXiv2017

The Dirichlet problem in a planar domain with two moderately close holes

We investigate a Dirichlet problem for the Laplace equation in a domain of $\mathbb{R}^2$ with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance $|ε_1|$ one from the other and each one of size $|ε_1ε_2|$. In such a domain, we introduce a Dirichlet problem and we denote by $u_{ε_1,ε_2}$ its solution. We show that the dependence of $u_{ε_1,ε_2}$ upon $(ε_1,ε_2)$ can be described in terms of real analytic maps of the pair $(ε_1,ε_2)$ defined in an open neighborhood of $(0,0)$ and of logarithmic functions of $ε_1$ and $ε_2$. Then we study the asymptotic behaviour of of $u_{ε_1,ε_2}$ as $ε_1$ and $ε_2$ tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between $ε_1$ and $ε_2$.

math.AP↗

arXiv2015

An existence result for a nonlinear transmission problems

Let $Ω^o$ and $Ω^i$ be open bounded subsets of $\mathbb{R}^n$ of class $C^{1,α}$ such that the closure of $Ω^i$ is contained in $Ω^o$. Let $f^o$ be a function in $C^{1,α}(\partialΩ^o)$ and let $F$ and $G$ be continuous functions from $\partialΩ^i\times\mathbb{R}$ to $\mathbb{R}$. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on $F$ and $G$ there exists at least one pair of continuous functions $(u^o, u^i)$ such that \[ \left\{ \begin{array}{ll} Δu^o=0&\text{in }Ω^o\setminus\mathrm{cl}Ω^i\,,\\ Δu^i=0&\text{in }Ω^i\,,\\ u^o(x)=f^o(x)&\text{for all }x\in\partialΩ^o\,,\\ u^o(x)=F(x,u^i(x))&\text{for all }x\in\partialΩ^i\,,\\ ν_{Ω^i}\cdot\nabla u^o(x)-ν_{Ω^i}\cdot\nabla u^i(x)=G(x,u^i(x))&\text{for all }x\in\partialΩ^i\,, \end{array} \right. \] where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions $(u^o, u^i)$ is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique.

math.AP↗