15 research outputs found
Oscillation of solutions of second-order nonlinear differential equations of Euler type
AbstractWe consider the nonlinear Euler differential equation t2x″+g(x)=0. Here g(x) satisfies xg(x)>0 for x≠0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t2x″+a(t)g(x)=0
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 from interior Current Density Imaging data
obtainable using MRI measurements. We only require knowledge of the magnitude
of one current generated by a given voltage on the boundary
. As previously shown, the corresponding voltage potential u in
is a minimizer of the weighted least gradient problem
with . In this paper we present an
alternating split Bregman algorithm for treating such least gradient problems,
for non-negative and . 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 from knowledge of its magnitude, and of the voltage on the
boundary. We then present several numerical experiments that illustrate the
convergence behavior of the proposed algorithm
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
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
A class of second order dilation invariant inequalities
We compute the best constants in some dilation invariant inequalities for the weighted L2-norms of -\u3b4u and 07u, with weights being powers of the distance from the origin