Weighted Sobolev spaces of radially symmetric functions

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.Comment: 38 page

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

We derive a singular version of the Sphere Covering Inequality which was recently introduced in Gui and Moradifam (Invent Math. https://doi.org/10.1007/s00222-018- 0820-2, 2018) suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce newuniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in Luo and Tian (Proc Am Math Soc 116(4):1119– 1129, 1992). Furthermore, we derive new symmetry results for the spherical Onsager vortex equation

A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. As previously shown, the corresponding voltage potential u in $\Omega$ is a minimizer of the weighted least gradient problem $$u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},$$ with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm

The Gel'fand problem for the biharmonic operator

We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains

On the homoclinic orbits of the generalized Liénard equations

AbstractIn this work we study the existence of homoclinic orbits of the planar system of Liénard type ẋ=1a(x)[h(y)−F(x)],ẏ=−a(x)g(x), where a(x)>0, for every x∈R, and h is strictly increasing, but it is not assumed that h(±∞)=±∞, h(y)≤my, or h(y)≥my. We present sufficient and necessary conditions for this system to have a positive orbit which starts at a point on the curve h(y)=F(x) and approaches the origin without intersecting the x-axis. The conditions obtained are very sharp. Our results extend the results presented by Hara and Yoneyama for this system with a(x)=1, and h(y)=y, and improve the results presented by Sugie

On the existence of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains

AbstractWe prove under quite general assumptions the existence of a bounded positive solution of the semilinear Schrödinger equation Δu+f(x,u)=0 in a two-dimensional exterior domain. Our results are independent of the behavior of f(x,u) when u is sufficiently small or sufficiently large and just require some knowledge about the nonlinearity f(x,u) for a≤u≤b, for some a,b>0. We obtain solutions with a prescribed positive lower bound

THE GENERALISED LIÉNARD EQUATIONS

AbstractIn this paper we present sufficient conditions for all trajectories of the system to cross the vertical isocline h(y) = F(x), which is very important in the global asymptotic stability of the origin, oscillation theory and existence of periodic solutions. Also we give sufficient conditions for all trajectories which start at a point on the curve h(y) = F(x), to cross the y-axis which is closely connected with the existence of homoclinic orbits, stability of the zero solution, oscillation theory and the centre problem. The obtained results extend and improve some of the authors' previous results and some other theorems in the literature.</jats:p

Intersection with the vertical isocline in the generalized Liénard equations

AbstractWe consider the generalized Liénard systemdxdt=1a(x)[h(y)−F(x)],dydt=−a(x)g(x), where a, F, g, and h are continuous functions on R and a(x)>0, for x∈R. Under the assumptions that the origin is a unique equilibrium, we study the problem whether all trajectories of this system intersect the vertical isocline h(y)=F(x), which is very important in the global asymptotic stability of the origin, oscillation theory, and existence of periodic solutions. Under quite general assumptions we obtain sufficient and necessary conditions which are very sharp. Our results extend the results of Villari and Zanolin, and Hara and Sugie for this system with h(y)=y, and a(x)=1 and improve the results presented by Sugie et al. and Gyllenberg and Ping