Boundedness of solutions to the Schrödinger equation under Neumann boundary conditions

AbstractWe deal with Neumann problems for Schrödinger type equations, with non-necessarily bounded potentials, in possibly irregular domains in Rn. Sharp balance conditions between the regularity of the domain and the integrability of the potential for any solution to be bounded are established. The regularity of the domain is described either through its isoperimetric function or its isocapacitary function. The integrability of the sole negative part of the potential plays a role, and is prescribed via its distribution function. The relevant conditions amount to the membership of the negative part of the potential to a Lorentz type space defined either in terms of the isoperimetric function, or of the isocapacitary function of the domain

On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions

By Birman and Skvortsov it is known that if \Omegasf is a planar curvilinear polygon with $n$ non-convex corners then the Laplace operator with domain H^2(\Omegasf)\cap H^1_0(\Omegasf) is a closed symmetric operator with deficiency indices $(n,n)$. Here we provide a Kre\u\i n-type resolvent formula for any self-adjoint extensions of such an operator, i.e. for the set of self-adjoint non-Friedrichs Dirichlet Laplacians on \Omegasf, and show that any element in this set is the norm resolvent limit of a suitable sequence of Friedrichs-Dirichlet Laplacians with $n$ point interactions.Comment: Slightly revised version. Accepted for publication in Journal of Functional Analysi

Hardy inequalities for Robin Laplacians

In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary conditions. We also study several extensions to non-convex and unbounded domains

On the best constants of Hardy inequality in Rn−k×(R+)k\mathbb{R}^{n-k}\times (\mathbb{R}_{+})^{k} and related improvements

We compute the explicit sharp constants of Hardy inequalities in the cone $\mathbb{R}_{k_+}^{n}:=\mathbb{R}^{n-k}\times (\mathbb{R}_{+})^{k}=\{(x_{1},...,x_{n})|x_{n-k+1}>0,...,x_{n}>0\}$ with $1\leq k\leq n$. Furthermore, the spherical harmonic decomposition is given for a function $u\in C^{\infty}_{0}(\mathbb{R}_{k_+}^{n})$. Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.Comment: 7 page

Fast cubature of volume potentials over rectangular domains by approximate approximations

In the present paper we study high-order cubature formulas for the computation of advection–diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order $\mathcal{O}(h^6)$ up to dimension $10^8$

Fast cubature of volume potentials over rectangular domains by approximate approximations

In the present paper we study high-order cubature formulas for the computation of advection–diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order $\mathcal{O}(h^6)$ up to dimension $10^8$

GLOBAL SOLUTIONS TO 2-D INHOMOGENEOUS NAVIER-STOKES SYSTEM WITH GENERAL VELOCITY

In this paper, we are concerned with the global wellposedness of 2-D density-dependent incompressible Navier-Stokes equations (1.1) with variable viscosity, in a critical functional frame- work which is invariant by the scaling of the equations and under a non-linear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques in [10] to prove the global existence of solutions to (1.1) with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of [10]