We are interested in algorithms for constructing surfaces $\Gamma$ of possibly small measure that separate a given domain $\Omega$ into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the $p$-Laplacians, $p\to 1$, under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients.

In this paper we introduce the notion of "$p$-dimensional rate of convergence" which generalizes the notion of rate of convergence introduced by V. Pták. Using this notion we give a generalization of the Induction Theorem of V. Pták, which may constitute a basis for the study of the iterative procedures of the form $X_{n+1}=F(x_{n-p+1},X_{n-p+2},...,x_n)$, $n=0,1,2,...$. As an illustration we apply these results to the study of the convergence of the secant method, obtaining sharp estimates for the errors at each step of the iterative procedure.

A general existence and uniqueness result of Picard-Lindelöf type is proved for ordinary differential equations in Fréchet spaces as an application of a generalized Nash-Moser implicit function theorem. Many examples show that the assumptions of the main result are natural. Applications are given for the Fréchet spaces $C^{\infty }\left(K\right)$, $S\left({ℝ}^N\right)$, $B\left(ℝR^N\right)$, $D_{L_1}\left({ℝ}^N\right)$, for Köthe sequence spaces, and for the general class of subbinomic Fréchet algebras.

Let $(𝒪,\sum ,F_{\infty })$ be an arithmetic ring of Krull dimension at most $1,S=\text{Spec}\left(𝒪\right)$ and $(𝒳\to S;{\sigma }_1,...,{\sigma }_n)$ a pointed stable curve. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(S\right)$. For every integer $k>0$, the invertible sheaf ${\omega }_{𝒳/S}^{k+1}(k{\sigma }_1+...+k{\sigma }_n)$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we define a Quillen type metric ${∥·∥}_Q$ on the determinant line ${\lambda }_{k+1}=\lambda {\omega }_{𝒳/S}^{k+1}$$(k...$

We prove an analog in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.

The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous compactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate...

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree...