Inverse problem for Sturm-Liouville operators on a curve

he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.

Rectifiable and nonrectifiable solution curves of half-linear differential systems

We study a geometric kind of asymptotic behaviour of every C1 solution of a class of nonautonomous systems of half-linear differential equations with continuous coefficients. We give necessary and sufficient conditions such that the image of every solution (solution curve) has finite length (rectifiable curve) and infinite length (nonrectifiable, possible fractal curve). A particular attention is paid to systems having attractive zero solution. The main results are proved by using a new result for the nonrectifiable plane curves.

Maximal spectral surfaces of revolution converge to a catenoid

We consider a maximization problem for eigenvalues of the Laplace–Beltrami operator on surfaces of revolution in R 3 with two prescribed boundary components. For every j , we show there is a surface Σ j that maximizes the j th Dirichlet eigenvalue. The maximizing surface has a meridian which is a rectifiable curve. If there is a catenoid which is the unique area minimizing surface with the prescribed boundary, then the eigenvalue maximizing surfaces of revolution converge to this catenoid.

Universal curves in the center problem for Abel differential equations

We study the center problem for the class ${\mathcal{E}}_{{\rm\Gamma}}$ of Abel differential equations $dv/dt=a_{1}v^{2}+a_{2}v^{3}$, $a_{1},a_{2}\in L^{\infty }([0,T])$, such that images of Lipschitz paths $\tilde{A}:=(\int _{0}^{\cdot }a_{1}(s)\,ds,\int _{0}^{\cdot }a_{2}(s)\,ds):[0,T]\rightarrow \mathbb{R}^{2}$ belong to a fixed compact rectifiable curve ${\rm\Gamma}$. Such a curve is said to be universal if whenever an equation in ${\mathcal{E}}_{{\rm\Gamma}}$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_{1},a_{2}$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with the product being the composition of series, explicit examples of universal curves, and approximation of Lipschitz triangulable curves by universal ones.

Intersection of Continua and Rectifiable Curves

We prove that for any non-degenerate continuum K ⊆ ℝd there exists a rectifiable curve such that its intersection with K has Hausdorff dimension 1. This answers a question of Kirchheim.