Given a smooth bounded domain $Ø\subseteq \R^2$, we consider the equation $\D v = 2 v_x \wedge v_y$ in $Ø$, where $v: Ø\to \R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron.
In this note we consider a heat trace expansion on a manifold with wedge-like singularity. We show that there are two terms in the expansion that contain information about the presence of the singularity, namely the logarithmic term $ct^{-1/2}\log t$ and the half power term $bt^{-1/2}$. We also give a geometric expression for $c$.
We obtain uniform asymptotics for deformed Airy kernel determinants, which arise in models of finite temperature free fermions and which characterize the narrow wedge solution of the Kardar-Parisi-Zhang equation. The asymptotics for the determinants yield uniform initial data for an associated family of solutions to the Korteweg-de Vries equation, and uniform lower tail asymptotics for the narrow wedge solution of the Kardar-Parisi-Zhang equation.
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \, x\in\mathbb{R},\,t>0,$ with a step-like boundary values: $q(x,t)\to 0$ as $x\to-\infty$ and $q(x,t)\to A$ as $x\to\infty$ for all $t\geq0$, where $A>0$ is a constant. The long-time asymptotics of the solution $q(x,t)$ of this problem along the rays $x/t=C\ne 0$ is presented in \cite{RS2}. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray $x=0$ than these rays with any nonzero constant $C$. …
The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of both the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangi…
By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation \[ -\{div} (|x|^θ\nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $Ω\subset \R^N \; (N \geq 2)$}, \leqno(1) \] where $p>1$, $θ, l\in\R^1$ with $N+θ>2$, $l-θ>-2$, and $Ω$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^σu(x)$, equation (1) can be rewritten as a nonlinear Schrödinger equation with Hardy potential $$-Δu=|x|^α|u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \{in $Ω\subset \R^N \;\; (N \geq 2)$}, \leqno{(2)}$$ where $p>1$, $α\in (-\infty, \infty)$ and $\ell \…
In this paper we consider the Dirichlet problem \begin{equation} \label{iniz} \begin{cases} -Δu = ρ^2 (e^{u} - e^{-u}) & \text{ in } Ω\\ u=0 & \text{ on } \partial Ω, \end{cases} \end{equation} where $ρ$ is a small parameter and $Ω$ is a $C^2$ bounded domain in $\mathbb{R}^2$. [1] proves the existence of a $m$-point blow-up solution $u_ρ$ jointly with its asymptotic behaviour. we compute the Morse index of $u_ρ$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.
Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.
We consider the Hénon equation \begin{equation}\label{alphab} -Δu = |x|^α|u|^{p-1}u \ \ \textrm{in} \ \ B^N, \quad u = 0 \ \ \textrm{on}\ \ \partial B^N, \tag{$P_α$} \end{equation} where $B^N\subset \mathbb{R}^N$ is the open unit ball centered at the origin, $N\geq 3$, $p>1$ and $α> 0$ is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation \[ -Δw = |w|^{p-1}w\quad \text{in}\ B^2,\quad w=0\quad \text{on}\ \partial B^2, \] where $B^2 \subset \mathbb{R}^2$ is the open unit ball, is the limit problem of \eqref{alphab}, as $α\to \infty$, in the framework o…
We compute the Morse index of nodal radial solutions to the Hénon problem \[\left\{\begin{array}{ll} -Δu = |x|^α|u|^{p-1} u \qquad & \text{ in } B, \newline u= 0 & \text{ on } \partial B, \end{array} \right. \] where $B$ stands for the unit ball in ${\mathbb R}^N$ in dimension $N\ge 3$, $α>0$ and $p$ is near at the threshold exponent for existence of solutions $p_α=\frac{N+2+2α}{N-2}$, obtaining that \begin{align*} m(u_p) & = m \sum\limits_{j=0}^{1+\left[α/{2}\right]} N_j \quad & \mbox{ if $α$ is not an even integer, or} \newline m(u_p)& = m\sum\limits_{j=0}^{ α/2} N_j + (m-1) N_{1+α/ 2} & \mb…
We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract}\left\{ \begin{array}{lr} -Δu= |u|^{p-1}u\qquad \mbox{ in }Ω\\ u=0\qquad\qquad\qquad\mbox{ on }\partial Ω\end{array} \right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $Ω$ is a smooth bounded symmetric domain of $\mathbb R^2$. We show that for families $(u_p)$ of sign-changing symmetric solutions of \eqref{problemAbstract} an upper bound on their Morse index implies concentration of the positive and negative part, $u_p^\pm$, at the same point, as $p\to+\infty$. Then an asymptotic analysis of $u_p^+$ an…
Let $Ω= \mathbb R^3 \setminus \bar{K}$, where $K$ is an open bounded domain with smooth boundary $Γ$. Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup related to Maxwell's equations in $Ω$ with dissipative boundary condition $ν\wedge (ν\wedge E)+ γ(x) (ν\wedge H) = 0, γ(x) > 0, \forall x \in Γ.$ We study the case when $γ(x) \neq 1, \: \forall x \in Γ,$ and we establish a Weyl formula for the counting function of the eigenvalues of $G_b$ in a polynomial neighbourhood of the negative real axis.
In this paper we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent $p$ is close to $1$. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle $2π/n$. By considerations based on the Morse index we see that, depending on the values of $α$ and $n$, such least energy solutions can be radial, or nonradial and…
In this work we study existence, asymptotic behaviour and stability properties of $O(m) times O(n)$ invariant solutions of the Allen-Cahn equation $Delta u+u(1-u^2)=0$ in $R^m times R^n$ with $m,n ge 2$, $m+n ge 8$. We exhibit four families of solutions whose nodal sets are smooth logarithmic corrections of the Lawson cone and with infinite Morse index. This work complements the study started by Pacard and Wei (about stable solutions) and by Agudelo, Kowalczyk and Rizzi (about 2 ended solutions).
We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function $ϕ(T)$ is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the prob…
We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the coordinate transformation of Euler-Lagrange type and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the Euler system, we derive a single second-order nonlinear elliptic equation for the potential fun…
We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey--Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family o…
In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi…
Let $U_h:\mathbb R^{d}\to \mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\,dt+\sqrt{2h}\,dB_t$$ in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-hΔ+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+hν$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $ν:\mathbb R^{d}\to \mathbb R^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac Vh}$ as an invariant measure for some smooth function $V:\mathbb{R}^d\rightarrow\mathb…
