The Properties of Solutions for a Generalized b-Family Equation with Peakons

This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y _t +u ^m+1 y _x +bu ^m u _x y=0, where b is a constant, $m\in \mathbb{N}$ , and we have the notation $y:= (1-\partial_{x}^{2}) u$ , which includes the famous Camassa–Holm equation, the Degasperis–Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces $H^{s}(\mathbb{R})$ with $s>\frac{3}{2}$ is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the $H^{1}(\mathbb{R})$ -norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces $H^{s}(\mathbb{R})$ with $1<s<\frac{3}{2}$ is established, under the assumption $u_{0}(x)\in H^{s}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$ . Finally, the global weak solution and peakon solution for the equation are also given.     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Alternating groups and flag-transitive triplanes

Let ${\mathcal{D}}$ be a nontrivial triplane, and G be a subgroup of the full automorphism group of ${\mathcal{D}}$ . In this paper we prove that if ${\mathcal{D}}$ is a triplane, ${G\leq Aut(\mathcal{D})}$ is flag-transitive, point-primitive and Soc(G) is an alternating group, then ${\mathcal{D}}$ is the projective space PG ₂(3, 2), and ${G\cong A_7}$ with the point stabiliser ${G_x\cong PSL_3(2)}$ .     

Locally homogeneous rigid geometric structures on surfaces

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ${\nabla}$ on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ${\nabla}$ be a unimodular real analytic affine connection on a real analytic compact connected surface M. If ${\nabla}$ is locally homogeneous on a nontrivial open set in M, we prove that ${\nabla}$ is locally homogeneous on all of M.     

The complex Monge–Ampère equation on compact K?hler manifolds

We consider the complex Monge–Ampère equation on a compact K?hler manifold (M, g) when the right hand side F has rather weak regularity. In particular we prove that estimate of ${\triangle \phi}$ and the gradient estimate hold when F is in ${W^{1, p_0}}$ for any p ₀?>?2n. As an application, we show that there exists a classical solution in ${W^{3, p_0}}$ for the complex Monge–Ampère equation when F is in ${W^{1, p_0}}$ .     

Q-Complete domains with corners in {\mathbb{P}^n} and extension of line bundles

We show that if a compact set X in ${\mathbb P^n}$ is laminated by holomorphic submanifolds of dimension q, then ${\mathbb P^n{\setminus}X}$ is (q + 1)-complete with corners. Consider a manifold U, q-complete with corners. Let ${\mathcal N}$ be a holomorphic line bundle in the complement of a compact in U. We study when ${\mathcal N}$ extends as a holomorphic line bundle in U. We give applications to the non existence of some Levi-flat foliations in open sets in ${\mathbb P^n}$ . The results apply in particular when U is a Stein manifold of dimension n ≥ 3, then every holomorphic line bundle in the complement of a compact extends holomorphically to U.     

New extremal domains for the first eigenvalue of the Laplacian in flat tori

We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds ${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$ with flat metric, for some T > 0. These domains are close to the cylinder-type domain ${B_1 \times \mathbb{R}{/}T\, \mathbb{Z}}$ , where B ₁ is the unit ball in ${\mathbb{R}^{n}}$ , they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boundary.     

On Some Holomorphic Cohomological Equations

Let Σ be a non compact Riemann surface and ${\gamma :\Sigma \longrightarrow \Sigma}$ an automorphism acting freely and properly such that the quotient M = Σ/γ is a non compact Riemann surface. Using the fact that Σ and M are Stein manifolds, we prove that, for any holomorphic function ${g : \Sigma \longrightarrow {\mathbb C}}$ and any ${\lambda \in {\mathbb C}}$ , there exists a holomorphic function ${f:\Sigma \longrightarrow {\mathbb C}}$ which is a solution of the holomorphic cohomological equation ${f \circ \gamma - \lambda f = g}$ .     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$