In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of mean curvature flow in $${\mathbb {R}}^{n+1}$$ all $$n\ge 3$$ : show that if a $$\{M_t\}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such $$M_t\cap B(x_0,\varepsilon )$$ is $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$...

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps $\mathbf{f}_{\varepsilon} \colon \mathbb{S}^{d-1} \to K$ which, for $\varepsilon > 0$, are defined on the sphere have pro...

In this work we apply the unfolding operator method to analyze asymptotic behavior of solutions $p$-Laplacian equation with Neumann boundary condition set in a bounded thin domain type $R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon g\left({x}/{\varepsilon^\alpha}\right)\right\rbrace$ where $g$ is positive periodic function. We study three cases $0<\alpha<1$...

We consider the stochastic Cahn-Hilliard equation with additive noise term $\varepsilon^\gamma g\, \dot{W}$ ($\gamma >0$) that scales interfacial width parameter $\varepsilon$. verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $\varepsilon^{-1}$ only enters polynomially; proof is based on higher-moment iterates, and (discrete) spectral es...

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}&gt;0, &gt;0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} +A_{11} +A_{12} \dot{x}_{2} =A_{20} +A_{21} +A_{22} \end{a...

Collisional energy loss of a fast particle in medium is mostly due to the polarization by electromagnetic fields particle. A small fraction carried away Cherenkov radiation. In chiral there an additional contribution induction anomalous current proportional magnetic field. It causes lose form We employ classical electrodynamics, adequate wide range energies, compute collisional homogenous plasm...

In 2013, Masson and Siljander determined a method to prove that the minimal p-weak upper gradient $$g_{f_\varepsilon }$$ for time mollification $$f_\varepsilon $$ , $$\varepsilon >0$$ of parabolic Newton–Sobolev function $$f\in L^p_\mathrm {loc}(0,\tau ;N^{1,p}_\mathrm {loc}(\Omega ))$$ with $$\tau $$\Omega open domain in doubling metric measure space $$(\mathbb {X},d,\mu )$$ supporting weak (1...

In this paper, we consider the following logarithmic Schrödinger equation$ \begin{equation*} -\varepsilon^2\Delta u + V(x)u = u\log u^2\ \ \text{in}\ {\mathbb R}^N, \end{equation*} $where $ \varepsilon&gt;0 $, N\ge 1 V(x)\in C({\mathbb R}) is a continuous potential which can be unbounded below. By variational methods and penalized idea, show that problem has family of solutions u_{\varepsilon} ...