We construct a new class of positive solutions for a classical semilinear elliptic problem in the plane which arise for instance as the standing-wave problem for the standard nonlinear Schrödinger equation or in nonlinear models in Turing's theory biological theory of pattern formation such as the Gray-Scott or Gierer-Meinhardt systems. The solutions we construct have the property that their energy over a ball of radius R grows linearly with R as R tends to infinity. These solutions are strongly related to the solutions of a Toda system.
Motivated by the study of non abelian Chern Simons vortices of non topological type in Gauge Field Theory, we analyse the solvability of planar Liouville systems of Toda type in presence of singular sources. We identify necessary and sufficient conditions on the "flux" pair which ensure the radial solvability of the system. Since the given system includes the (integrable) 2 X 2 Toda system as a particular case, thus we recover the existence result available in this case. Our method relies on a blow-up analysis, which even in the radial setting, takes new turns compared with the single equation…
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
We consider the singular Liouville equation and the Henon-Lane-Emden problem on simply connected planar domains. We show that any solution to each problem must satisfy a uniform bound on the mass. The same results applies to some systems and more general non-linearities. The proofs are based on the Riemann mapping theorem and a Pohozaev-type identity.
In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of the boundary and the normal derivative of f on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex domains.
This paper is concerned with the asymptotic stability of planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic-elliptic coupled system of the radiating gas in half space. We show that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. These results are proved by the standard $L^2$-energy method. Moreover, we prove that the solution $(u,q)$ converges to the corresponding planar stationary solution at the rate $t^{-α/2-1/4}$ for non-degenerate case, and $t^{-1…
This paper studies the asymptotic stability of solution to an initial-boundary value problem for a hyperbolic-elliptic coupled system on two-dimensional half space, where the data on the boundary and at the far field are prescribed as $u_-$ and $u_+$. We show that the solution to the problem converges to the corresponding planar rarefaction wave for $0\le u_-<u_+$ as time tends to infinity under smallness assumptions on the initial perturbation and wave strength. These results are based on the analysis of div-curl decomposition, the standard $L^2$-energy method, $L^1$-estimate, and the monoton…
A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is proportional to the gradient of the domain's Green function with a fixed singularity on the interior. In this paper we review the Laplacian growth model and its key underlying assumptions, so that we may consider a generalization to so-called elliptic growth, wherein the Green function is replaced with that of a more general elliptic operator--this models, for example, inhomogeneities in the underlying plane. In this paper we continue the development o…
In this paper we consider the so-called Toda System in planar domains under Dirichlet boundary condition. We show the existence of continua of solutions for which one component is blowing up at a certain number of points. The proofs use singular perturbation methods.
This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in $L^p$ norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with strictly increasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improvement version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension…
We study positive solutions $u_p$ of the nonlinear Neumann elliptic problem $Δu =u$ in $Ω$, $\partial u/\partialν= |u|^{p-1}u$ on $\partialΩ$, where $Ω$ is a bounded open smooth domain in $\mathbb{R}^2$. We investigate the asymptotic behavior of families of solutions $u_p$ satisfying an energy bound condition when the exponent $p$ is getting large. Inspired by the work of Davila-del Pino-Musso \cite{DavilaDM}, we prove that $u_p$ is developing $m$ peaks $x_i\in\partial Ω$, in the sense $u_p^p/\int_{\partial Ω}u_p^p$ approaches the sum of $m$ Dirac masses at the boundary and we determine the lo…
We consider the equation $-Δu= |x|^α|u|^{p-1}u$ for any $α\geq 0$, either in $\mathbb R^2$ or in the unit ball $B$ of $\mathbb R^2$ centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as $p\rightarrow +\infty$ of all the radial solutions to these problems. In particular, we show that there is no uniform a priori bound (in $p$) for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the recently shown fact that positive solutions have uniform a priori bounds for $α=0$ and Dirichlet boundary …
We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.
This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.
A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case
Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）；Dipartimento di Matematica, Universita degli Studi di Bari Aldo Moro（巴里阿尔多·莫罗大学数学系）；Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine（乌迪内大学数学、信息学和物理科学系）
We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,θ) = Rg(θ)$, where $(R,θ)$ are plane polar coordinates and $g: \mathbb{R}^{2} \to \mathbb{R}^{m}$, $m \geq 2$. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals $I(u) = \int_{B}W(x,\nabla u(x))\,dx$, where $D_{F}W(x,F)$ behaves like $\frac{1}{|x|}$ as $|x| \to 0$ and $W$ satisfies an ellipticity condition. Such solutions cannot exist when $|x|D_{F}W(x,F) \to 0$ as $|x| \to 0$, so the condition is op…
Let $Ω$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$ \begin{cases} -Δ\upsilon= e^{\upsilon}-sϕ_1-4παδ_p-h(x)\,\,\,\, \,\textrm{in}\,\,\,\,\,Ω,\\[2mm] \upsilon=0 \quad\quad\quad\quad\quad\quad \qquad\qquad\quad\quad\,\,\,\, \textrm{on}\,\ \,\partialΩ, \end{cases} $$ where $s>0$ is a large parameter, $h\in C^{0,γ}(\overlineΩ)$, $p\inΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $δ_p$ denotes the Dirac measure supported at point $p$ and $ϕ_1$ is a positive first eigenfunction of the problem $-Δϕ=λϕ$ under Dirichlet boundary conditio…
In this paper, we consider the following elliptic Toda system associated to a general simple Lie algebra with multiple singular sources \begin{equation*} \begin{cases} -Δw_i=\sum_{j=1}^na_{i,j}e^{2w_j}+2π\sum_{\ell=1}^mβ_{i,\ell}δ_{p_\ell} \quad&\mbox{in}\quad\mathbb{R}^2,\\ \\ w_i(x)=-2\log|x|+O(1)~\mbox{as}~|x|\to\infty,\quad &i=1,\cdots,n, \end{cases} \end{equation*} where $β_{i,\ell}\in[0,1)$. Under some suitable assumption on $β_{i,\ell}$ we establish the existence and non-existence results. This paper generalizes Luo-Tian's [19] and Hyder-Lin-Wei's [10] results to the general Toda system…
Let $Ω$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2α}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $ν$ denotes the outer unit normal vector to $\partialΩ$, $q\in\overlineΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth fun…
The Riemann-Hilbert boundary value problem is studied for a class of planar complex vector fields $L$ in a simply connected open set $\Om\subset\R^2$. The first integrals of $L$ are used to reduce the problem into a collection of classical Riemann-Hilbert problems with discontinuous data.
Comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated in the last decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved.
A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.
We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.
We study nonlinear time-asymptotic stability of small--amplitude planar Lax shocks in a model consisting of a system of multi--dimensional conservation laws coupled with an elliptic system. Such a model can be found in context of dynamics of a gas in presence of radiation. Our main result asserts that the standard uniform Evans stability condition implies nonlinear stability. The main analysis is based on the earlier developments by Zumbrun for multi-dimensional viscous shock waves and by Lattanzio-Mascia-Nguyen-Plaza-Zumbrun for one--dimensional radiative shock profiles.
We address the well-posedness of the 2D (Euler)-Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich's type data, which gives a positive answer to part of the questions raised in 2011 [Lai-Pan-Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 739-760]. Our analysis on the the polygonal-like domains essentially relies on the recent elliptic regularity results for such domains proved in 2013 [Bardos-Plinio-Temam, The Euler equations in planar nonsmooth convex…
We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $\mathbb{C}$ that depend smoothly on a real parameter $λ\in [0,1]$. We derive sharp regularity properties of the solutions in all variables, including the parameter $λ$. More specifically we show that the solution and its derivatives are continuous in all variables, and the Hölder norms of the space variables are bounded uniformly in $λ$.
