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 condition in $Ω$. If $p$ is a strict local maximum point of $ϕ_1$, we show that such a problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity $\int_Ωe^{\upsilon_s}\rightarrow8π(m+1+α)ϕ_1(p)$ as $s\rightarrow+\infty$.
We revisit the following nonlinear critical elliptic equation \begin{equation*} -Δu+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about the periodicity of bubbling solutions. Here we try to investigate some related problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely many bubbling solutions of the problem above. This local uniqueness result implies that some bubbling solutions preserve the sy…
In this work we study the existence of nodal solutions for the problem $$ -Δu = λu e^{u^2+|u|^p} \text{ in }Ω, \; u = 0 \text{ on }\partial Ω, $$ where $Ω\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $Ω$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $Ω$ is an arbitrary domain and $λ$ is small enough. As far as we know, this is the first construction of sign-c…
We investigate bubbling solutions for the nonlocal equation \[ A_Ω^s u =u^p,\ u >0 \quad \mbox{in } Ω, \] under homogeneous Dirichlet conditions, where $Ω$ is a bounded and smooth domain. The operator $A_Ω^s$ stands for two types of nonlocal operators that we treat in a unified way: either the spectral fractional Laplacian or the restricted fractional Laplacian. In both cases $s \in (0,1)$ and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe $u=0$ on $\partial Ω$ and for the restricted fractional Laplacian, we prescribe $u=0$ on $\mathbb R^n \setminus…
A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main applications of the compactness theorem are discussed. First, we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential. Second, we prove an Alexandrov-type theorem for crystalline isoperimetric problems.
This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem: \begin{equation*} \begin{cases} -Δu_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -Δu_2=K_2(y)u_1^{q}, \ y\in \mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5, p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. At first, under suitable conditions on $K_1,K_2$ and the certain range of the exponents $p,q$, we construct an unbounded s…
In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger-Moser exponential growth: \begin{equation*}\begin{cases} -dΔu_d+u_d=u_d(e^{u^2_{d}}-1)\ \ \mbox{in}\ Ω,\\ \frac{\partial u_d}{\partialν}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial Ω,\\ \end{cases}\end{equa…
We study an elliptic equation related to the Moser-Trudinger inequality on a compact Riemann surface $(S,g)$, $$ Δ_g u+λ\Biggl(ue^{u^2}-{1\over |S|} \int_S ue^{u^2} dv_g\Biggl)=0,\quad\text{in $S$},\qquad \int_S u\,dv_g=0, $$ where $λ>0$ is a small parameter, $|S|$ is the area of $S$, $Δ_g$ is the Laplace-Beltrami operator and $dv_g$ is the area element. Given any integer $k\geq 1$, under general conditions on $S$ we find a bubbling solution $u_λ$ which blows up at exactly $k$ points in $S$, as $λ\to0$. When $S$ is a flat two-torus in rectangular form, we find that either seven or nine familie…
Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear effect. One observes a strict splitting in space between an almost linear regime and a dominantly non-linear regime which has two major characteristics : it requires a quantized amount of energy and arise…
In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (Ann. Inst. H. Poincare Anal. Non Lineaire, 1991) and Rey (J. Funct. Anal., 1990).
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…
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.
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…
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 the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system \begin{equation}\label{e051} \left\{ \begin{split} &Δu_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=4π\sum_{i=1}^{2k}δ_{p_{1,i}}\\ &Δu_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=4π\sum_{i=1}^{2k}δ_{p_{2,i}} \end{split} \text{ in }\quad Ω\right., \end{equation} where $k\geq 1$ and $Ω$ is a flat tours in $\mathbb{R}^2$. It continues the joint work with Zhang\cite{HZ-2015}, where we obtained the necessary conditions for the existence of bubbling solutions of Liouville type. Under nearly necessar…
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 …
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…
The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$Δu+h_1e^u-h_2e^{-u}=0 \quad \mathrm{in}~B_1\subset\mathbb{R}^2.$$ In particular we get that the blow-up values are multiple of $8π.$ It generalizes the result of Jost, Wang, Ye and Zhou \cite{jwyz} where the extra assumption $h_1 = h_2$ is crucially used.
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.
Motivated by a challenging expectation of Rivière (2011), in the recent interesting work of deLongueville-Gastel (2019), de Longueville and Gastel proposed the following geometrical even order elliptic system \begin{equation*} Δ^{m}u=\sum_{l=0}^{m-1}Δ^{l}\left\langle V_{l},du\right\rangle +\sum_{l=0}^{m-2}Δ^{l}δ\left(w_{l}du\right)\qquad \text{ in } B^{2m}\label{eq: Longue-Gastel system} \end{equation*} which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order p…
Let $Ω$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partialΩ$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.
We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary conditions. Our method is based on an interplay between quasiconformal mappings, elliptic operators and composition operators on Sobolev spaces.
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.
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition in non-Lipschitz domains $\widetildeΩ \subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Neumann boundary condition in non-Lipschitz domains $Ω\subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.
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.
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.
In convex planar domains, given an initial vorticity with one sign, we study the regularity and geometric properties of the dynamically stable solutions to the Euler equations in the coadjoint orbit of the initial vorticity. These flows have elliptic stagnation points. Under some nondegeneracy conditions on the data, we show they are Holder continuous and have convex level curves. We also give a detailed description for the set of stagnation points. If the initial vorticity has nice level set topology, these stable solutions are in the L^\infty-strong closure of the coadjoint orbit. We also de…
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 compute the Morse index of $1$-spike solutions of the semilinear elliptic problem \begin{equation}\label{abstr} \tag{$\mathcal P_p$} \begin{cases} -Δu= u^p & \text{in $Ω$} \\ u=0 & \text{on $\partialΩ$} \\ u>0 & \text{in $Ω$.} \end{cases} \end{equation} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large. When $Ω$ is convex, our result, combined with the characterization in [22], a result in [41] and with recent uniform estimates in \cite{Sirakov}, gives the uniqueness of the solution to \eqref{abstr}, for $p$ large. This proves, in dimension two and for…
In this paper, we study steady vortex patch solutions to the incompressible Euler equations in a planar bounded domain $D$. Let $ψ_0$ be the solution of the elliptic problem $-Δψ_{0} =1$ in $D$; $ψ_0=0$ on $\partial D$. We prove that for any finite collection of isolated maximum points of $ψ_0$, say $\{x_1,\cdot\cdot\cdot,x_k\},$ and any $k$-tuple $\vecκ=(κ_1,\cdot,\cdot,\cdot,κ_k)$ with $κ_i>0$ and $|\vecκ|:=\sum_{i=1}^kκ_i<<1,$ there exists a steady solution of the Euler equations such that the vorticity has the form $ω^{\vecκ}=1-I_{\cup_{i=1}^k A^{\vecκ}_i}$, where $I$ denotes the character…
We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish back-projection type inversion formulas that recover any initial data with support in $\Om$ modulo an explicitly computed smoothing integral operator $\K_\Om$. For circular and elliptical domains the operator $\K_\Om$ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtaine…
In this survey we formulate our results on different forms of maximum principles for linear elliptic equations and systems. We start with necessary and sufficient conditions for validity of the classical maximum modulus principle for solutions of second order strongly elliptic systems. This principle holds under rather heavy restrictions on the coefficients of the systems, for instance, it fails for the Stokes and Lamé systems. Next, we turn to sharp constants in more general maximum principles due to S. Agmon and C. Miranda. We consider higher order elliptic equations, Stokes and Lamé systems…
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 $λ$.
We find a complete characterization for sets of isotropic conductivities with stable recovery in the $L^2$ norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the modulus of continuity of the coefficients involved and their ellipticity bound.
Let $D_t$, $t \in [0,1]$ be an arbitrary 1-parameter family of Dirac type operators on a two-dimensional disk with $m-1$ holes. Suppose that all operators $D_t$ have the same symbol, and that $D_1$ is conjugate to $D_0$ by a scalar gauge transformation. Suppose that all operators $D_t$ are considered with the same locally elliptic boundary condition, given by a vector bundle over the boundary. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of the holes in the d…
