Related papers: A singularly perturbed Dirichlet problem for the P…

Let $\Omega$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in \Omega$ and that $\mathbb{R}^n \setminus \mathrm{cl}\Omega$ is connected. Then we take $q_{11},..., q_{nn}\in ]0,+\infty[$ and $p \in…

We consider a sufficiently regular bounded open connected subset $\Omega$ of $\mathbb{R}^n$ such that $0 \in \Omega$ and such that $\mathbb{R}^n \setminus \cl\Omega$ is connected. Then we choose a point $w \in ]0,1[^n$. If $\epsilon$ is a…

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $\epsilon>0$ proportional to the radius of the holes and a map $\phi$,…

We take an open regular domain $\Omega$ in $\mathbb{R}^n$ with $n\ge 3$. We introduce a pair of positive parameters $\epsilon_1$ and $\epsilon_2$ and we set $\epsilon\equiv(\epsilon_1,\epsilon_2)$. Then we define the perforated domain…

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by $\epsilon$ > 0, and is proportional to the distance between neighbouring perforations. In the…

We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $R^{n}$ indexed by a small parameter $\epsilon$. The domains depend on $\epsilon$ only within a ball of radius proportional to $\epsilon$ and,…

We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $\Omega_\varepsilon$ with holes concentrated at subsets of a domain $\Omega$ non-periodically. We show $u^\varepsilon$ converges to a solution…

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain…

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\Omega\subset\mathbb{R}^n$ is…

We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain \begin{equation*} \left\{ \begin{array}{ll} \Delta u +\rho\left(V_1(x)e^{u}- V_2(x)e^{-\tau u}\right)=0 &\text{in }…

In this paper we establish $W^{1,p}$ estimates for solutions $u_\varepsilon$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain $\Omega_{\varepsilon, \eta}$ in…

Let n\ge 3. Let \Omega^i and \Omega^o be open bounded connected subsets of R^n containing the origin. Let \epsilon_0>0 be such that \Omega^o contains the closure of \epsilon\Omega^i for all \epsilon\in]-\epsilon_0,\epsilon_0[. Then, for a…

In this paper we study the Dirichlet problem for Laplace's equation in a domain $\omega_{\epsilon, \eta}$ perforated periodically with small holes in $\mathbb{R}^d$, where $\epsilon$ represents the scale of the minimal distances between…

This paper studies the Dirichlet problem for Laplace's equation in a domain $\Omega_{\varepsilon, \eta}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $\eta$ the ratio…

It is shown that if the sequence $(p_j(x))$ increases uniformly to $p(x)$ in a bounded, smooth domain $\Omega$, then the sequence $(u_i)$ of solutions to the Dirichlet problem for the $p_i(x)$-Laplacian with fixed boundary datum $\varphi$…

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $\Gamma$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the…

We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }\Omega\\ v=\varphi \text{ on }\partial\Omega. \end{cases} $$…

In this paper we analyze nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int_{B} J(x-y) (u(y) - u(x)) dy$ with $x$ in a perforated domain $\Omega^\epsilon \subset \Omega$. Here $J$ is a…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value…

In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, $\varepsilon>0$, tends to zero. We consider pairwise interaction energies satisfying $p$-growth conditions, $p<d$, $d$ being the…