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David Ruiz

Publications and source records attributed to David Ruiz.

At least 37 records · Page 2Linked to original sources
arXiv2019

Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime

In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Maris in 2013. However, such result is valid only in dimension 3 and higher. In this paper we first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3. With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3, recovering the aforementioned result by Maris. The planar case turns out to be more difficult and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions.

math.AP↗
arXiv2019

Conformal metrics with prescribed Gaussian and geodesic curvatures

We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a conformal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange energy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have diverging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up analysis around a sequence of points different from local maxima; the use of holomorphic domain-variations; and Morse-index estimates.

math.AP↗
arXiv2017

Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials

In this paper we consider a mean field problem on a compact surface with conical singularities. This problem appears in the Gaussian curvature prescription problem in Geometry, and also in the Electroweak Theory and in the abelian Chern-Simons-Higgs model in Physics. In this paper we focus on the case of sign-changing potentials, and we give results on compactness, existence and multiplicity of solutions.

math.AP↗
arXiv2016

Solutions to overdetermined elliptic problems in nontrivial exterior domains

In this paper we construct nontrivial exterior domains $Ω\subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -Δu +u -u^p=0,\ u >0 & \mbox{in }\; Ω, {1mm] \ u= 0 & \mbox{on }\; \partial Ω, [1mm] \ \frac{\partial u}{\partial ν} = \mbox{cte} & \mbox{on }\; \partial Ω, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain.

math.AP↗
arXiv2016

Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as stationary Hartree equation or Schrödinger--Newton equation) \[ -Δu + u = (I_α\star |u|^p) |u|^{p - 2}u. \] Here $I_α$ stands for the Riesz potential of order $α\in (0,N)$, and $\frac{N - 2}{N + α} < \frac{1}{p} \le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when $α$ is either close to $0$ or close to $N$.

math.AP↗
arXiv2015

A general existence result for the Toda system on compact surfaces

In this paper we consider the Toda system of equations on a compact surface, which is motivated by the study of models in non-abelian Chern-Simons theory. We prove a general existence result using variational methods. The same analysis applies to a mean field equation which arises in fluid dynamics.

math.AP↗
arXiv2015

A rigidity result for overdetermined elliptic problems in the plane

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $Ω\subset \mathbb{R}^2$ a $C^{1,α}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem $$ \left\{\begin{array} {ll} Δu + f(u) = 0 & \mbox{in }\; Ω \\ u= 0\, \, \, , \, \, \, \frac{\partial u}{\partial \vecν}=1 &\mbox{on }\; \partial Ω\end{array}\right. $$ we prove that $Ω$ is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.

math.AP↗
arXiv2015

Standing waves for a gauged nonlinear Schrödinger equation with a vortex point

This paper is motivated by a gauged Schrödinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational structure. We will study the global behavior of that functional, extending known results for the regular case.

math.AP↗
arXiv2014

Asymmetric blow-up for the SU(3) Toda System

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.

math.AP↗
arXiv2014

Prescribing the Gaussian curvature in a subdomain of S^2 with Neumann boundary condition

In this paper we study the problem of prescribing the Gaussian curvature under a conformal change of the metric. We are concerned with the problem posed on a subdomain of the 2-sphere under Neumann boundary conditions of the conformal factor. If the area of the subdomain is greater than 2π, the associated energy functional is no longer bounded from below. We treat this case by using min-max techniques, giving a new existence result that generalizes and unifies previous work on the argument.

math.AP↗
arXiv2013

On the Leray-Schauder degree of the Toda system on compact surfaces

In this paper we consider the so-called Toda system of equations on a compact surface. In particular, we discuss the parity of the Leray-Schauder degree of that problem. Our main tool is a theorem of Krasnoselskii and Zabreiko on the degree of maps symmetric with respect to a subspace. This result yields new existence results as well as a new proof of previous results in literature.

math.AP↗
arXiv2013

Boundary concentration of a Gauged Nonlinear Schrodinger Equation

This paper is motivated by a gauged Schrodinger equation in dimension 2 including the so-called Chern-Simons term. The radially symmetric case leads to an elliptic problem with a nonlocal defocusing term, in competition with a local focusing nonlinearity. In this work we pose the equations in a ball under homogeneous Dirichlet boundary conditions. By using singular perturbation arguments we prove existence of solutions for large values of the radius. Those solutions are located close to the boundary and the limit profile is given.

math.AP↗
arXiv2013

A Variational Analysis of a Gauged Nonlinear Schrödinger Equation

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - Δu(x) + \left(ω+ \frac{h^2(|x|)}{|x|^2} + \int_{|x|}^{+\infty} \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x), $$ where $$ h(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, ds. $$ This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in(1,3)$, the functional may be bounded from below or not, depending on $ω$. Quite surprisingly, the threshold value for $ω$ is explicit. From this study we prove existence and non-existence of positive solutions.

math.AP↗
arXiv2012

A note on the uniformity of the constant in the Poincaré inequality

The classical Poincaré inequality establishes that for any bounded regular domain $Ω\subset \R^N$ there exists a constant $C=C(Ω)>0$ such that $$ \int_Ω |u|^2\, dx \leq C \int_Ω |\nabla u|^2\, dx \ \ \forall u \in H^1(Ω),\ \int_Ω u(x) \, dx=0.$$ In this note we show that $C$ can be taken independently of $Ω$ when $Ω$ is in a certain class of domains. Our result generalizes previous results in this direction.

math.AP↗
arXiv2011

A variational Analysis of the Toda System on Compact Surfaces

In this paper we consider the Toda system of equations on a compact surface. We will give existence results by using variational methods in a non coercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u_1, u_2.

math.AP↗