Related papers: Linear Stability Implies Nonlinear Stability for F…

We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of…

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a K\"ahler-Einstein manifold to more general K\"ahler manifolds including a Fano manifold equipped with a…

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard…

The Barashenkov-Bogdan-Zhanlav solitons $u_\pm$ for the forced NLS/Lugiato-Lefever model on the line are considered. While the instability of $u_+$ was established in the original paper, \cite{B1}, the analogous question for $u_-$ was only…

We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show…

Let $\Omega$ be a domain in $R^d$ and $d_\Gamma$ the Euclidean distance to the boundary $\Gamma$. We investigate whether the weighted Hardy inequality \[ \|d_\Gamma^{\delta/2-1}\varphi\|_2\leq…

The nonlinear Schr\"odinger equation NLSE(p, \beta), -iu_t=-u_{xx}+\beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space \Lp^2(\mT). For p\leq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that…

We prove area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a corresponding upper bound for the area in terms of the charges obtained recently by…

We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal…

In this study, we examine the linear stability of an axisymmetric Taylor bubble moving steadily in a flowing liquid enclosed in a circular tube. Linearisation is performed about axisymmetric base states obtained in Part I of this study by…

This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $\tau(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2}…

The well-known Faber-Krahn theorem states that the ball has the lowest first Dirichlet eigenvalue among all domains of the same volume in $\mathbb{R}^n$. Leydold (Geom. Funct. Anal, 1997) gave the discrete version of Faber-Krahn inequality…

Let $\mathbf{a}$ be a quadratic form associated with a Schr\"odinger operator $L=-\nabla\cdot(A\nabla)+V$ on a domain $\Omega\subset \mathbb{R}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(\Omega)$, then either there is $W>0$ such…

We consider a family of regularized defocusing nonlinear Schrodinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by…

Consider the nonlinear parabolic equation in the form $$ u_t-{\rm div} \mathbf{a}(D u,x,t)={\rm div}\,(|F|^{p-2}F) \quad \text{in} \quad \Omega\times(0,T), $$ where $T>0$ and $\Omega$ is a Reifenberg domain. We suppose that the nonlinearity…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$. For $\epsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-\Delta u=\frac{1}{\epsilon^2}(1-|u|^2)u$ in $\Omega$ such that on $\partial \Omega$ u…

We study the $L^{\infty}$ stability of the 2D Navier-Stokes equations with a viscosity-dependent Navier boundary condition around shear profiles which are linearly unstable for the Euler equation. The dependence from the viscosity is given…

The Hardy constant of a simply connected domain $\Omega\subset\R^2$ is the best constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \;, u\in C^{\infty}_c(\Omega). \]…

A stability version of the Blaschke-Santal\'o inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial…