arXiv subjects

David Ruiz

Publications and source records attributed to David Ruiz.

At least 19 recordsLinked to original sources

arXiv2026

Conformal Metrics on the Disk with Prescribed Negative Gaussian Curvature and Boundary Geodesic Curvature

We study the problem of prescribing the Gaussian curvature on the disk and the geodesic curvature on its boundary via a conformal change of the metric. In this paper the case of negative Gaussian curvature is treated, a regime for which the bubbling behavior of approximate solutions is not so well understood. This is due to the possible appearance of blow-up solutions with diverging length and area. We give an existence result under assumptions on the curvatures which are somewhat natural, in view of some obstructions inherent to the problem. Our strategy is variational and relies on the study of certain families of approximated problems. By performing a refined blow-up analysis for solutions with bounded Morse index, we conclude compactness.

math.AP↗

arXiv2025

Uniformly rotating Euler flows with compactly supported velocity

For any positive integer $k$, we prove the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D incompressible Euler equations with compact spatial support. These solutions, which can be chosen to be small perturbations of radial flows, are the first example of smooth rotating flows with finite energy which are not locally radial. We also prove new rigidity results for rotating solutions which show that the geometric structure of these flows is severely constrained.

math.AP↗

arXiv2025

Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip

This paper focuses on establishing the existence of a class of steady solutions, termed least total curvature solutions, to the incompressible Euler system in a strip. The solutions obtained in this paper complement the least total curvature solutions already known. Our approach employs a minimization procedure to identify a monotone heteroclinic solution for a conveniently chosen semilinear elliptic PDE. This method also enables us to construct positive and monotone (and consequently stable) solutions to semilinear elliptic PDEs with non-convex superlevel sets in a strip domain. This can be regarded as a negative answer to a generalized problem raised in [27].

math.AP↗

arXiv2025

Rigidity results for finite energy solutions to the stationary 2D Euler equations

In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the corresponding stream function solves an autonomous semilinear elliptic equation. Under some extra conditions on the vorticity near infinity we can also prove that the streamlines are concentric circles. The proofs include several energy estimates on the behavior of the stream function at infinity, as well as an adaptation of the continuous Steiner symmetrization to our setting.

math.AP↗

arXiv2024

Monotone heteroclinic solutions to semilinear PDEs in cylinders and applications

In this paper we show the existence of strictly monotone heteroclinic type solutions of semilinear elliptic equations in cylinders. The motivation of this construction is twofold: first, it implies the existence of an entire bounded solution of a semilinear equation without critical points which is not one-dimensional. Second, this gives an example of a bounded stationary solution for the 2D Euler equations without stagnation points which is not a shear flow, completing previous results of Hamel and Nadirashvili. The proof uses a minimization technique together with a truncation argument, and a limit procedure.

math.AP↗

arXiv2024

Smooth nonradial stationary Euler flows on the plane with compact support

We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class $C^k(\mathbb{R}^2)$ which are not locally radial. The proof uses a degree-theory-based bifurcation argument which hinges on three key ingredients: a novel approach to stationary Euler flows through elliptic equations with non-autonomous nonlinearities; a set of sharp regularity estimates for the linearized operator, which involves a potential that blows up as the inverse square of the distance to the boundary of the support; and overcoming a serious problem of loss of derivatives by the introduction of anisotropic weighted functional spaces between which the linearized operator is Fredholm.

math.AP↗

arXiv2024

Nontrivial solutions to the relative overdetermined torsion problem in a cylinder

Given a bounded regular domain $\omega \subset \mathbb{R}^{N-1}$ and the half-cylinder $\Sigma = \omega \times (0,+\infty)$, we consider the relative overdetermined torsion problem in $\Sigma$, i.e. \[\left\{ \begin{array}{ll} \Delta {u}+1=0 &\mbox{in $\Omega$},\newline \partial_\eta u = 0 &\mbox{on $\widetilde \Gamma_\Omega$},\newline u=0 &\mbox{on $\Gamma_\Omega$},\newline \partial_{\nu}u =c &\mbox{on $\Gamma_\Omega$}. \end{array} \right. \] where $\Omega \subset \Sigma$, $\Gamma_\Omega = \partial \Omega \cap \Sigma$, $\widetilde \Gamma_\Omega = \partial \Omega \setminus \Gamma_\Omega$, $\nu$ is the outer unit normal vector on $\Gamma_\Omega$ and $\eta$ is the outer unit normal vector on $\widetilde \Gamma_\Omega$. We build nontrivial solutions to this problem in domains $\Omega$ that are the hypograph of certain nonconstant functions $v : \overline{\omega} \to (0, + \infty)$. Such solutions can be reflected with respect to $\omega$, giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for any generic base $\omega$.

math.AP↗

arXiv2024

Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature

This paper deals with the question of prescribing the Gaussian curvature on a disk and the geodesic curvature of its boundary by means of a conformal deformation of the metric. We restrict ourselves to a symmetric setting in which the Gaussian curvature is negative, and we are able to give general existence results. Our approach is variational, and solutions will be searched as critical points of an associated functional. The proofs use a perturbation argument via the monotonicity trick of Struwe, together with a blow-up analysis and Morse index estimates. We also give a nonexistence result that shows that, to some extent, the assumptions required for existence are necessary.

math.DG↗

arXiv2023

A Schiffer-type problem for annuli with applications to stationary planar Euler flows

If on a smooth bounded domain $\Omega\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $\Omega$ be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function $u$ is allowed to take a different constant value on each connected component of $\partial \Omega$ yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains $\Omega$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $\Omega$ and of the bounded component of the complement $\mathbb{R}^2\backslash\overline{\Omega}$ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial.

math.AP↗

arXiv2023

Modica type estimates and curvature results for overdetermined elliptic problems

In this paper, we establish a Modica type estimate on bounded solutions to the overdetermined elliptic problem \begin{equation*} \begin{cases} \Delta u+f(u) =0& \mbox{in $\Omega$, }\\ u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=-\kappa &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^{n},n\geq 2$. As we will see, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. We will also discuss the equality case. From such estimates we will deduce information about the curvature of $\partial \Omega$ under a certain condition on $\kappa$ and $f$. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.

math.AP↗

arXiv2022

Nonsymmetric sign-changing solutions to overdetermined elliptic problems in bounded domains

In 1971 J. Serrin proved that, given a smooth bounded domain $\Omega \subset \mathbb{R}^N$ and $u$ a positive solution of the problem: \begin{equation*} \begin{array}{ll} -\Delta u = f(u) &\mbox{in $\Omega$, } u =0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u =\mbox{constant} &\mbox{on $\partial\Omega$, } \end{array} \end{equation*} then $\Omega$ is necessarily a ball and $u$ is radially symmetric. In this paper we prove that the positivity of $u$ is necessary in that symmetry result. In fact we find a sign-changing solution to that problem for a $C^2$ function $f(u)$ in a bounded domain $\Omega$ different from a ball. The proof uses a local bifurcation argument, based on the study of the associated linearized operator.

math.AP↗

arXiv2022

Overdetermined elliptic problems in nontrivial contractible domains of the sphere

In this paper, we prove the existence of nontrivial contractible domains $\Omega\subset\mathbb{S}^{d}$, $d\geq2$, such that the overdetermined elliptic problem \begin{equation*} \begin{cases} -\varepsilon\Delta_{g} u +u-u^{p}=0 &\mbox{in $\Omega$, } u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=\mbox{constant} &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} admits a positive solution. Here $\Delta_{g}$ is the Laplace-Beltrami operator in the unit sphere $\mathbb{S}^{d}$ with respect to the canonical round metric $g$, $\varepsilon>0$ is a small real parameter and $1 1$ if $d=2$). These domains are perturbations of $\mathbb{S}^{d}\setminus D,$ where $D$ is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.

math.AP↗

arXiv2022

Qualitative Properties of Singular Solutions to the Fractional Yamabe Problem

In this paper we are interested in the qualitative properties of the solutions to the fractional Yamabe problem in $\mathbb{R}^n$ which present an isolated singularity. In particular, we prove that the Morse index of any such solution is infinity. The proof uses a Emden Fowler type transformation, so that we can pass to a nonlocal 1D problem posed in $\mathbb{R}$.

math.AP↗

arXiv2022

Symmetry results for compactly supported steady solutions of the 2D Euler equations

In this paper we prove symmetry of compactly supported steady solutions of the 2D Euler equations. Assuming that $\Omega = \{x \in \mathbb{R}^2:\ u(x) \neq 0\}$ is an annular domain, we prove that the streamlines of the flow are circular. We are also able to remove the topological condition on $\Omega$ if we impose regularity and nondegeneracy assumptions on $u$ at $\partial \Omega$. The proof uses that the corresponding stream function solves an elliptic semilinear problem $-\Delta \phi = f(\phi)$ with $\nabla \phi=0$ at the boundary. One of the main difficulties in our study is that $f$ is not Lipschitz continuous near the boundary values. However, $f(\phi)$ vanishes at the boundary values and then we can apply a local symmetry result of F. Brock to conclude. In the case $\partial_{\nu} u \neq 0$ at $\partial \Omega$ this argument is not possible. In this case we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of $f$. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, Hopf lemma and Serrin corner lemma for elliptic linear operators with singular coefficients.

math.AP↗

arXiv2021

Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori

In this paper we consider traveling waves for the Gross-Pitaevskii equation which are T-periodic in each variable. We prove that if T is large enough, there exists a solution as a global minimizer of the corresponding action functional. In the subsonic case, we can use variational methods to prove the existence of a mountain-pass solution. Moreover, we show that for small T the problem admits only constant solutions.

math.AP↗

arXiv2021

Conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results under conditions that involve $K$ and $H$, where $H$ denotes the harmonic extension of $h$. The proof is based on the computation of the Leray-Schauder degree in a compact setting.

math.AP↗

arXiv2020

LEAPME: Learning-based Property Matching with Embeddings

Data integration tasks such as the creation and extension of knowledge graphs involve the fusion of heterogeneous entities from many sources. Matching and fusion of such entities require to also match and combine their properties (attributes). However, previous schema matching approaches mostly focus on two sources only and often rely on simple similarity measurements. They thus face problems in challenging use cases such as the integration of heterogeneous product entities from many sources. We therefore present a new machine learning-based property matching approach called LEAPME (LEArning-based Property Matching with Embeddings) that utilizes numerous features of both property names and instance values. The approach heavily makes use of word embeddings to better utilize the domain-specific semantics of both property names and instance values. The use of supervised machine learning helps exploit the predictive power of word embeddings. Our comparative evaluation against five baselines for several multi-source datasets with real-world data shows the high effectiveness of LEAPME. We also show that our approach is even effective when training data from another domain (transfer learning) is used.

cs.DB↗

arXiv2020

Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

This paper is concerned with the compactness of metrics of the disk with prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence of metrics and give a precise description of its asymptotic behavior. In particular, the metrics blow-up at a unique point on the boundary and we are able to give necessary conditions on its location. It turns out that such conditions depend locally on the Gaussian curvatures but they depend on the geodesic curvatures in a nonlocal way. This is a novelty with respect to the classical Nirenberg problem where the blow-up conditions are local, and this new aspect is driven by the boundary condition.

math.AP↗