arXiv subjects

D. Ruiz

Publications and source records attributed to D. Ruiz.

5 recordsLinked to original sources

arXiv2021

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the overdetermined elliptic problem \begin{equation*} \begin{cases} \Delta u +f(u)=0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=\mbox{constant} &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +\infty) \to \mathbb{R}$. Roughly speaking, we only ask that the Dirichlet problem in $B$ admits a nondegenerate solution. The proof uses a local bifurcation argument.

math.AP↗

arXiv2021

Conformal metrics with prescribed scalar and mean curvature

We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur with profiles of hyperbolic balls or horospheres, and hence one may lose either pointwise estimates on the conformal factor or the total conformal volume. We can sometimes prevent them using integral estimates, Pohozaev identities and domain-variations of different types.

math.AP↗

arXiv2019

On numerical solution of full rank linear systems

Matrices can be augmented by adding additional columns such that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. This augmented system can then be solved efficiently by a sum of projections onto these subspaces. The equivalence to the original linear system is ensured by adding additional rows to the matrix in a specific form. The resulting solution method is known as the augmented block Cimmino method. Here this method is extended to full rank underdetermined systems and to overdetermined systems. In the latter case, rows of the matrix, not columns, must be suitably augmented. The article presents an analysis of these methods.

math.NA↗

arXiv2018

Prescribing Gaussian and geodesic curvature on disks

In this paper we consider the problem of prescribing the Gaussian and geodesic curvature on a disk and its boundary, respectively, via a conformal change of the metric. This leads us to a Liouville-type equation with a nonlinear Neumann boundary condition. We address the question of existence by setting the problem in a variational framework which seems to be completely new in the literature. We are able to find minimizers under symmetry assumptions.

math.AP↗

arXiv2018

The augmented Block Cimmino algorithm revisited

In this paper we replay the definitions, constructions and results from [SIAM J. Sci. Comp., 37(3)(2015), A1248-A1269] by completing and developing some of them to inconsistent least squares problems.

math.NA↗