Related papers: Homogenization of diffusion processes with singula…

Let $\Omega \subset \mathbb{R}^N$ be an open set. In this work we consider solutions of the following gradient elliptic system \[ -\text{div}(A(x)\nabla u_{i,\beta}) = f_i(x,u_{i,\beta}) + a(x)\beta |u_{i, \beta}|^{\gamma -1}u_{i, \beta}…

We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \\ u> 0, &x\in…

In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex…

We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…

We study the evolution of a passive scalar subject to molecular diffusion and advected by an incompressible velocity field on a 2D bounded domain. The velocity field is $u = \nabla^\perp H$, where H is an autonomous Hamiltonian whose level…

We continue the program initiated in a previous work, of applying integro-differential methods to Neumann Homogenization problems. We target the case of linear periodic equations with a singular drift, which includes (with some regularity…

We study how the resolvent-family of a diffusion behaves, as thedrift grows to infinity. The limit turns out to be a selfadjoint pseudo-resolvent.After reduction of the underlying Hilbert-space, this pseudo-resolvent becomesa resolvent to a…

In this paper, we study equations driven by a non-local integrodifferential operator $\mathcal{L}_K$ with homogeneous Dirichlet boundary conditions. More precisely, we study the problem \[ \begin{aligned} &- \mathcal{L}_K u + V(x)u =…

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \[ \left\{ \begin{array}{c} u_{t} \ - \ (…

This paper deals with semilinear elliptic problems of the type \[ \left\{ \begin{array}{ll} -\Delta u+\alpha(x)u= \beta (x)|u|^{p-1}u \quad \hbox{in }\mathbb{R}^N, u(x)>0\quad\hbox{in } \mathbb{R}^N, \qquad u \in H^1(\mathbb{R}^N),…

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $\Xi^{g,{\bf a}} $ and bi-Laplacian fields $\Xi^{b,{\bf a}}$. They can be characterized as follows: for $f=\delta$ the solution $u$ of $\nabla \cdot…

The purpose of this article is two-fold. First, we investigate the inequality $$ -\Delta u+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we…

We consider mass concentration properties of Laplace eigenfunctions $\varphi_\lambda$, that is, smooth functions satisfying the equation $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$, on a smooth closed Riemannian manifold. Using a…

This paper is a generalization of the author's previous work [14]. We extend the argument [14] for any uniformly elliptic operator in divergence form $\mathcal{L}u=-div(A(x)\nabla u)$, more precisely, we study a fractional type degenerate…

We study the asymptotic behavior as $p\to\infty$ of the Gelfand problem \[ -\Delta_{p} u=\lambda\,e^{u}\ \textrm{in}\ \Omega\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partial\Omega. \] Under an appropriate rescaling on $u$ and $\lambda$,…

This paper provides sharp quantitative and constructive estimates of nonnegative solutions $u(t,x)\geq 0$ to the nonlinear fractional diffusion equation, $$\partial_t u +{\mathcal L} F(u)=0,$$ also known as filtration equation, posed in a…

We develop unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$…

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schr\"odinger equation $$ \left\{ \begin{array}{ll} -\Delta u = \lambda u + f(u)\quad & \text{in…

In the present work we shall describe and apply the techniques of the Renormalization Group - based in data rescaling and operator renormalizing - and of Homogenization - that substitutes, in a certain limit, a periodically inhomogeneous…