No abstract

Several authors have designed variants of Newton’s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton’s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane.

We prove a characterization of Hardy’s inequality in Sobolev–Slobodeckiĭ spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona Kinnunen & Korte for standard Sobolev spaces. The proof is based on variational methods.

We prove a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the p−Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no regularity assumptions, both on the set and the functions. We then show that such a comparison principle can be applied to prove: uniqueness of solutions; sharp pointwise estimates for positive solutions in convex sets; localization estimates for maximum points and sharp geometric estimates for generalized principal frequencies in convex sets. Contents 1. Introduction 1.1. The Lane-Emden equation 1.2. Main results 1.3. Some comments 1.4. Plan of the paper 2. Preliminaries 2.1. Notation 2.2. Sobolev spaces 2.3. Weak solutions 2.4. Hidden convexity 3. Some properties of the energy functional 4. A comparison principle 5. Applications to geometric estimates 5.1. Solutions of the Lane-Emden equation 5.2. Localization of maximum points 5.3. Generalized principal frequencies Appendix A. Quantified convexity of power functions Appendix B. Some subsolutions Appendix C. Asymptotics of the positive solution in a slab-type sequence References