arXiv subjects

Pilar Herreros

Publications and source records attributed to Pilar Herreros.

12 recordsLinked to original sources

arXiv2023

Multiplicity results for bound state solutions of a semilinear equation

We consider the problem of multiplicity and uniqueness of radial solutions of a nonlinear elliptic equation of the form \begin{eqnarray*} \begin{gathered} Δu +f(u)=0,\quad x\in \mathbb{R}^N, N\geq 2, \lim\limits_{|x|\to\infty}u(x)=0. \end{gathered} \end{eqnarray*} where $f$ is a prescribed function, satisfying appropriate conditions. This paper is a review of recent developments on the multiplicity of solutions, with a special emphasis on the effect of the mid-range behavior of the nonlinearity $f$ over the existence and multiplicity of solutions.

math.AP↗

arXiv2023

A new approach to the study of elliptic semilinear equations

In this paper we define a new operator $J$ for the study of $$ Δu +f(u)=0,\quad x\in R ^N, N> 2. $$ Using $J$ we can easily see some qualitative properties of the solutions, for example we can determine how many times $u$ changes sign, which are the values of the local maxima and minima, and where $u$ changes concavity. We also use this functional to construct nonlinearities $f$ such that this problem has at least two bound state solutions that change sign $j$ times, for $j=1,\dots,k-1$. And another $f$ such that this problem has a unique ground state solution, and at least two bound state solutions that change sign one time.

math.AP↗

arXiv2022

Multiplicity results for ground state solutions of a semilinear equation via abrupt changes in magnitude of the nonlinearity

Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$Δu +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground state solutions.

math.AP↗

arXiv2020

On the uniqueness of solutions of a semilinear equation in an annulus

We establish the uniqueness of positive radial solutions of $$\begin{cases} Δu +f(u)=0,\quad x\in A \\ u(x) =0 \quad x\in \partial A \end{cases} $$ where $A:=A_{a,b}=\{ x\in {\mathbb R}^n : a<|x| 0$ for all $s>0$, or has one zero at $B>0$, is non positive and not identically 0 in $(0,B)$ and it is positive in $(B,\infty)$.

math.AP↗

arXiv2019

On the uniqueness of bound state solutions of a semilinear equation with weights

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to $$ {div}\big(\mathsf A\,\nabla v\big)+\mathsf B\,f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,$$ $n>2$, where $\mathsf A$ and $\mathsf B$ are two positive, radial, smooth functions defined on $\mathbb R^n\setminus\{0\}$. We assume that the nonlinearity $f\in C(-c,c)$, $0 0$, is non positive and not identically 0 in $(0,b)$, positive in $(b,c)$, and is differentiable in $(0,c)$.

math.AP↗

arXiv2015

Multiplicity results for sign changing bound state solutions of a semilinear equation

In this paper we give conditions on $f$ so that problem $$ Δu +f(u)=0,\quad x\in \mathbb{R}^N, N\ge 2, $$ has at least two radial bound state solutions with any prescribed number of zeros, and such that $u(0)$ belongs to a specific subinterval of $(0,\infty)$. This property will allow us to give conditions on $f$ so that this problem has at least any given number of radial solutions having a prescribed number of zeros.

math.AP↗

arXiv2014

Regularity of Mediatrices in Surfaces

For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves exhibiting at most countably many singularities, with finite total angular deficiency.

math.DG↗

arXiv2013

On the uniqueness of the limit for an asymptotically autonomous semilinear equation on R^N

We consider a parabolic equation of the form u_t=Δu +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions, each globally defined and nonnegative bounded solution u converges to a single steady state.

math.AP↗

arXiv2011

Lens rigidity with trapped geodesics in two dimensions

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat Möbius strip are determined by their lens data. We also see by example that the flat Möbius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid.

math.DG↗

arXiv2010

Scattering Boundary Rigidity in the Presence of a Magnetic Field

It has been shown in \cite{DPSU} that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in \cite{DPSU} to regions with the same scattering data, where only one is known apriori to be simple. We will then use this results to resolve a local version of a question by Robert Bryant. That is, we show that a surface of constant curvature can not be modified in a small region while keeping all the curves of some fixed constant geodesic curvatures closed.

math.DG↗

arXiv2009

Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field

Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology, metric, and magnetic field are uniquely determined by the scattering relation of the magnetic geodesic flow, measured at the boundary. Conjugate points are allowed with minor restrictions. In exchange for the real-analytic assumption, prior results in boundary rigidity are greatly generalized.

math.DG↗

arXiv2007

Blocking: New examples and properties of products

We say that a pair of points x and y is secure if there exist a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena both in the manifold and the billiard table setting. As an approach to this, we study if the product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security that imposes that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we give an example of a compact C^1 surface that contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, as well as billiard table examples.

math.DG↗