Let $\Delta u={\Sigma }_i\frac{{\partial }^2}{{\partial }_{x_i}^2}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^2}{\partial x_i\partial x_j}u+{\Sigma }_i{b}_i\frac{\partial }{\partial x_i}u+cu$ be elliptic operators with Hölder continuous coefficients on a bounded domain $\Omega \subset {\mathbf{R}}^n$ of class $C^{1,1}$. There is a constant $c\>0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that $$c^{-1}{G}_{\Delta }^{\Omega }\le G_L^{\Omega }\le c{G}_{\Delta }^{\Omega }\text{on}\Omega \times \Omega ,$$ where ${\gamma }_{\Delta }^{\Omega }$ (resp. $G_L^{\Omega }$) are the Green functions of $\Delta$ (resp. $L$) on $\Omega$.

We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions ${\chi }_a(t,x)$ form a family of evolving regular curves in ${ℝ}^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We consider curves that are small regular perturbations of ${\chi }_a(t_0,x)$ for a fixed time $t_0$. In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of...

We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system $i\partial ₜu+\Delta u=\lambda ₁\left|u\right|⁴u+\lambda ₂b₁u{v}_{x₁}$, $-\Delta v=b₂{\left(\right|u\left|²\right)}_{x₁}$. Under certain conditions on λ₁ and λ₂, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281-1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794-1797].

We study local reflections ${\varphi }_{\sigma }$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if ${\varphi }_{\sigma }$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ${\varphi }_{\sigma }$ is harmonic for all geodesies $\sigma$.

We study local reflections ${\varphi }_{\sigma }$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if ${\varphi }_{\sigma }$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ${\varphi }_{\sigma }$ is harmonic for all geodesies $\sigma$.

The paper deals with very weak solutions $u\in \theta +W_0^{1,r}\left(\Omega \right)$, $max\{1,p-1\}<r<p<n$, to boundary value problems of the $p$-harmonic equation $$\left\{\begin{array}{cc}-\text{div}\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=0,\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=\theta \left(x\right),\hfill & x\in \partial \Omega .\hfill \end{array}\right.(*)$$ We show that, under the assumption $\theta \in W^{1,q}\left(\Omega \right)$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$u\in \left\{\begin{array}{cc}\theta +L_{\mathrm{weak}}^{q^{*}}\left(\Omega \right)\hfill & \text{for}q<n,\hfill \\ \theta +L_{\mathrm{weak}}^{\tau }\left(\Omega \right)\hfill & \text{for}q=n\text{and}\text{any}\tau <\infty ,\hfill \\ \theta +L^{\infty }\left(\Omega \right)\hfill & \text{for}q>n,\hfill \end{array}\right.$$ provided that $r$ is sufficiently close to $p$.

We consider the classical three-dimensional motion in a potential which is the sum of $n$ attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the $n$ centres, we find a universal behaviour for all energies $E$ above a positive threshold. Whereas for $n=1$ there are no bounded orbits, and for $n=2$ there is just one closed orbit, for $n\ge 3$ the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of...

Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_{\pm }(\lambda ;H₀+rV,H₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including...

A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation ${\Delta }^{p}f=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:\Omega \to {ℂ}^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f=(f₁,...,f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $({ℝ}^m,e^{{\left|x\right|}^2/2(m-2)}/d{s}_0^2)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_0$ is the Euclidean metric in ${ℝ}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems...

Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty }$ manifold $X$. Let $p_m$ be the principal symbol and assume that $\Sigma =p_m^{-1}\left(0\right)$ is an involutive $C^{\infty }$ sub-manifold of $T^{*}X\setminus 0$, satisfying a certain transversality condition. We assume that $p_m$ vanishes exactly to order $M$ on $\Sigma$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M=2$). Suppose that a Levi condition is valid for the lower order symbols. If $u\in {𝒟}^{\text{'}}\left(X\right)$, $Pu\in C^{\infty }\left(X\right)$, then...

We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{\text{loc}}^{1,1}({ℝ}^N;{ℝ}^d)$ and in the space $B{V}_{\text{loc}}({ℝ}^N;{ℝ}^d)$ of functions of bounded variation.

Some properties of the functions of the form $v\left(x\right)={\sum }_{i=0}^m{\left|x\right|}^i{h}_i\left(x\right)$ in ℝⁿ, n ≥ 2, where each $h_i$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

A "matching" method for planar harmonic functions is exhibited, using an elliptic adjoint equation $L^{\star }u=0$.

We prove that if $f:Z^d\to R$ is harmonic and there exists a polynomial $W:Z^d\to R$ such that f + W is nonnegative, then f is a polynomial.

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\&gt;0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \&gt;0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition $$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|\&lt;r}|x-y{|}^{2-n}v\left(y\right)dy=0$$ for all compact $K\subset \Omega$, where if $n=2$, we replace $|x-y{|}^{2-n}$ by $-log|x-y|$. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$ $$\frac{\left|u\right(x\left)\right|}{|x{|}^N}\le C{\int }_{{\mathbf{R}}^3}|x-y{|}^{-1}\frac{\left|\Delta u\right(y\left)\right|}{|y{|}^N}dy\text{for}\text{a.e.}x\text{in}{\mathbf{R}}^3.$$