Title: Generic properties of minimal surfaces.
Speaker: Antonio Alarcón (Universidad de Granada).
Abstract:
We shall discuss some properties of minimal surfaces in Euclidean space which hold generically in Baire category sense. Based on joint work with Francisco J. López.
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Title: The blessing of smoothness and outlier-free spline spaces in isogeometric analysis.
Speaker: Hendrik Speleers (University of Rome Tor Vergata).
Abstract:
Isogeometric analysis is a well-established paradigm to improve interoperability between geometric modeling and numerical simulation. It is based on smooth spline representations and shows important advantages over classical \(\mathcal{C}^0\) finite element analysis. In particular, the higher smoothness enables a higher accuracy per degree of freedom. This superior performance has been observed numerically since long time, and recently is supported theoretically by error estimates with constants that are explicit, not only in the mesh size, but also in the polynomial degree and the smoothness. Moreover, the isogeometric approach based on maximally smooth spline spaces over uniform grids turns out to be an excellent choice for addressing eigenvalue problems: it gives a very good approximation of the full spectrum, except for a very small portion of spurious outliers.
In this seminar we review some recent results on explicit error estimates for approximation with highly smooth splines. These estimates are sharp or close to sharp in several interesting cases and are actually good enough to cover convergence to eigenfunctions of classical differential operators under so-called k-refinement. We also discuss how subspaces of maximally smooth splines, which are optimal in the sense of Kolmogorov n-widths, can be selected to make outlier-free isogeometric discretizations. The seminar is based on joint work with Carla Manni and Espen Sande.
Título: The impact of terms of order 1 in Dirichlet problems with discontinuous coefficients.
Conferenciante: Lucio Boccardo (Sapienza Università di Roma, Istituto Lombardo).
Resumen:
Descargar resumen aquí.
Toda la información puede encontrarse en:
Seminario de Ecuaciones Diferenciales.
Title: Applications of minimal surfaces to geometry and topology of 3-manifolds, II.
Speaker: Yevgeny Liokumovich, University of Toronto.
Abstract:
Minimal surfaces are a key tool in the study of geometry and topology of 3-manifolds. In the first talk I will give an overview of several problems that can be resolved using the existence and regularity theory of minimal surfaces:
In the second talk, I will describe some of the proofs in more detail.
The seminar will also be streamed at
Zoom virtual room 1.
For further information, visit:
Title: Applications of minimal surfaces to geometry and topology of 3-manifolds, I.
Speaker: Yevgeny Liokumovich, University of Toronto.
Abstract:
Minimal surfaces are a key tool in the study of geometry and topology of 3-manifolds. In the first talk I will give an overview of several problems that can be resolved using the existence and regularity theory of minimal surfaces:
In the second talk, I will describe some of the proofs in more detail.
The seminar will also be streamed at
Zoom virtual room 1.
For further information, visit:
Title: Submersions, Lie groups and Killing Vector Fields to Study Translating Solitons of the Mean Curvature Flow.
Speaker: Miguel Ortega Titos (UGR).
Abstract:
Graphical translating solitons of the mean curvature flow (translators) can be studied in many ambient manifolds. Some authors have used rotationally invariant settings to work with an ODE. We recall the associated PDE in a semi-riemannian setting for product manifolds. We apply these ideas to study translators in Minkowski space which are invariant by the orthogonal group and the orthochronus group. One step further, when the ambient manifold admits a non-unit, non-vanishing Killing vector field, it can be seen as a warped product. Then, it is possible to make a similar study. We obtain the corresponding PDE and make a general theory for the reduction to an ODE. We will show some new examples in this way.
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Título: Rigidity results for the capillary overdetermined problem.
Conferenciante: Yuanyuan Lian (Universidad de Granada).
Resumen (Descargar resumen aquí):
In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem:
\begin{equation}
\begin{cases}
\text{div} \left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) + f(u) = 0 &\text{ in } \Omega, \\
u = 0 &\text{ on } \partial\Omega, \\
\partial_{\nu} u = \kappa &\text{ on } \partial\Omega,
\end{cases}
\tag{1}
\end{equation}
where \( f \) is a given \( C^1 \) function in \( \mathbb{R} \), \( \nu \) is the exterior unit normal, \( \kappa \) is a constant and \( \Omega \subset \mathbb{R}^n \) is a \( C^1 \) domain. Our main theorem states that if \( n = 2 \), \( \kappa \neq 0 \), \( \partial\Omega \) is unbounded and connected, \( |\nabla u| \) is bounded, and there exists a nonpositive primitive \( F \) of \( f \) such that \(F(0) \geq (1 + \kappa^2)^{-\frac{1}{2}} - 1\), then \( \Omega \) must be a half-plane and \( u \) is a parallel solution. In other words, under our assumptions, if a capillary graph has the property that its mean curvature depends only on the height, then it is the graph of a one-dimensional function. We also prove the boundedness of the gradient of solutions of (1) when \( f'(u) 0 \). Moreover, we study a Modica type estimate for the overdetermined problem (1) that allows us to prove that, unless \( \Omega \) is a half-space, the mean curvature of \( \partial\Omega \) is strictly negative under the assumption that \( \kappa \neq 0 \) and there exists a nonpositive primitive \( F \) of \( f \) such that \( F(0) \geq (1 + \kappa^2)^{-\frac{1}{2}} - 1\). Our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where \( f \) is linear.
Contact: yuanyuanlian@correo.ugr.es
Toda la información puede encontrarse en:
Seminario de Ecuaciones Diferenciales.
Title: Rigidity and symmetry results of some elliptic problems in
punctured domains.
Speaker: Francesco Esposito (Università della Calabria).
Abstract:
The aim of this talk is to show qualitative properties of solutions to some
nonlinear elliptic problems. The presentation is divided into two parts.
In the first part, we will focus on semilinear elliptic problems. Specifically, we
will present a recent rigidity result, inspired by a seminal paper of J. Serrin,
but considering the case of singular solutions to some semilinear overdetermined elliptic problems in punctured domains. The proofs rely on a refined application of the moving planes method, as developed by Serrin and Gidas-Ni-Nirenberg.
The second part of the talk will be addressed to quasilinear elliptic problems
and potential developments.
This talk is based on some joint papers in collaboration with S. Biagi, A.
Farina, L. Montoro, B. Sciunzi, N. Soave and E. Vecchi.
Download abstract here.
All the information can be found at:
Seminario de Ecuaciones Diferenciales.