Perturbations near the vertex of a plane sector and a 3D cone, Matteo Dalla Riva (University of Tulsa, USA), 13 maggio 2021, ore 12:30, (online, Zoom)
Abstract: The Functional Analytic Approach (FAA) is a method to study boundary value problems in singularly perturbed domains. It is somehow alternative to the more classical techniques of Asymptotic Analysis and, typically, it allows a representation of the solutions in terms of real analytic functions of the perturbation parameter. But there are problems for which we cannot expect real analyticity. For example, this is the case of a Poisson equation in a sector or cone domain with a perturbation pattern that approaches the vertex. We will see how the FAA can be modified to study these problems.
The results presented are part of a joint work with M. Costabel, M. Dauge, and P. Musolino.
TBA, Francesco Ferraresso (Universität Bern, CH), 17 dicembre 2019, ore 12:20, aula seminari 430
Abstract: TBA.
A distributional approach to fractional Sobolev spaces and fractional, Giorgio Stefani (SNS, IT), 11 Dicembre 2019, ore 14:30 aula 2BC30
Abstract: In a recent paper in collaboration with G. E. Comi, we have introduced a new notion of fractional $\alpha$-variation for all $\alpha\in(0,1)$ via a distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In this talk, after a brief summary on the relevant features of our fractional BV space, I will present some recent results in collaboration with E. Brué, M. Calzi and G. E. Comi about the asymptotic behaviour of the fractional $\alpha$-variation as the parameter $\alpha$ tends to 0 and 1. We are able to prove natural analogues of the asymptotic results by Davila-Maz'ya-Shaposhnikova via new interpolation inequalities. In addition, when $\alpha$ tends to 1, we are also able to prove the $\Gamma$-convergence of the fractional variation to the De Giorgi's variation, in perfect analogy with the well-known $\Gamma$-convergence result by Ambrosio-De Philippis-Martinazzi.
Biharmonic eigenvalues on annuli and rectangles, Davide Buoso (EPFL), 30 Ottobre 2019, ore 14:30 aula 2BC30
Abstract: In this talk we discuss some nodal properties and shape optimization results for the eigenvalues of the Dirichlet Bilaplacian and of the Buckling problem. In particular, we focus our attention on two types of planar domains: annuli and rectangles. Annuli are good to study the nodal domains of eigenfunctions because they can be written explicitly in terms of Bessel functions, and we show that, quite unexpectedly, the two problems have completely different behaviors in terms of nodal domains. However, in both cases the annulus (of given area) that realizes the minimum first eigenvalue can be shown to be the ball in both cases. On the other hand, the situation for rectangles is reversed: any eigenfunction will present oscillations near the corners, but the shape optimization problem is highly non-trivial and we conjecture that it is solved by the square. We will also compare these results with the analogs for the Dirichlet Laplacian to show how the needed techniques change when passing from second-order problems to higher-order ones. Based on joint works with P. Freitas and E. Parini.
Existence, uniqueness, and regularity properties of the solutions of a nonlinear transmission problem, Matteo Dalla Riva (University of Tulsa), 18 Luglio 2019, ore 14:45 aula 2AB45
Abstract: We consider a nonlinear contact problem that arises in the study of composite structures glued together by thermo-active materials. By an approach based on boundary integral equations and on the Schauder fixed-point theorem we can prove that the problem has solutions; however, such solution may not be locally unique and may be also very irregular. For example, we might have solutions that are not in Hs for any s > 1/2. The results presented are obtained in collaboration with B. Luczak (the University of Tulsa, US), G. Mishuris (Aberystwyth University, UK), R. Molinarolo (Aberystwyth University, UK), and P. Musolino (Università degli Studi di Padova, Italy).
Global Analysis of Locally Symmetric Spaces with Indefinite metric, Toshiyuki Kobayashi (Tokyo University), 3 giugno 2019, ore 10:30 aula 1BC45
Abstract: The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure. In this talk, I plan to discuss two topics.
Global geometry: Existence problem of compact manifolds modeled locally on homogeneous spaces, and their deformation theory.
Spectral analysis: Construction of periodic eigenfunctions for the (indefinite) Laplacian, and stability question of eigenvalues under deformation of geometric structure.
Funzioni con primitiva, Jürgen Appell (Würzburg University), 30 aprile 2019, ore 16:45 aula 2AB40
Abstract: In ogni corso di Analisi 1 si impara che ogni funzione continua $f : [a, b] \to \mathbb{R}$ ha una primitiva (cioè è la derivata $f = F'$ di un’altra funzione $F$ ), ed ogni funzione con primitiva ha la proprietà del valor intermedio (cioè soddisfa $f ([\alpha,\beta]) \supseteq [f (\alpha), f (\beta)] ∪ [f(\alpha),f(\beta)]$ per ogni intervallo $[\alpha,\beta] \subseteq [a,b]$).
A questo punto studentesse più brave a volte pongono due domande:
• Si può vedere che una funzione ha una primitiva, senza calcolarla?
• Esistono funzioni con la proprietà del valor intermedio, ma senza primitiva?
Inoltre, visto che la somma $f + g$ di due funzioni $f$ e $g$ con primitiva ha anche una primitiva (che è banale!), si pongono in maniera naturale le seguenti ulteriori domande:
• Se $f$ e $g$ hanno una primitiva, anche il prodotto $fg$ ha una primitiva?
• Se $f$ e $g$ hanno una primitiva, anche la composizione $g \circ f$ ha una primitiva?
Nella conferenza diamo la risposta completa a tutte le domande, basata in parte su ricerche recenti. Invece di discutere le dimostrazioni (che sono tutt’altro che banali) presentiamo una serie di esempi illustrativi. La presentazione dei risultati non richiede conoscenze specialistiche e pertanto è accessibile anche a tutti gli studenti.
A class of Hausdorff - Berezin operators on the unit disc, Alexey Karapetyants (Southern Federal University and Don State Technical University of Rostov on Don), 27 marzo 2019, ore 15:30 aula 1BC45
Abstract: We introduce and study the class of Hausdorff-Berezin operators on the unit disc in the Lebesgue p-spaces with Haar measure. We discuss certain algebraic properties of such operators, and also give sufficient, and, in some cases necessary boundedness conditions for such operators. Joint work with Profs. K. Zhu and S. Samko.
Boundary blow-up solutions to semilinear elliptic equations with Hardy potential, Vitaly Moroz (Swansea University), 21 febbraio 2019, ore 12.00 aula 2AB40
Abstract: Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of the potential affects the existence of a "large solutions": if Hardy parameter is small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence, existence and precise asymptotics for different types of solutions will be given. This is based on joint works with Catherine Bandle, Moshe Marcus and Wolfgang Reichel.
The spectrum of the Dirichlet-to-Neumann operator for submanifolds of R^n with fixed boundary, Bruno Colbois (Université de Neuchâtel), 12 febbraio 2019, ore 12:15, aula 1BC5
Abstract: We consider the spectrum of the Dirichlet-to-Neumann operator on a family of compact submanifold of $\mathbb{R}^n$ with fixed boundary. I will first recall briefly the problem of finding geometric bounds for the spectrum of a Laplace-type operator. For the Dirichlet-to-Neumann operator, in the particular case of revolution submanifolds, we can obtain very precise bounds for all the eigenvalues. In the general case, I will explain how to find upper bounds depending only on the volume of the submanifold and also how to construct examples with arbitrarily small eigenvalues. I will profit of the talk to present a couple of open questions concerning the spectrum of the Dirichlet-to-Neumann operator that are easy to formulate. They are a good illustration of the kind of questions that are of interest in this area.
Minimizers for energies related to dislocations, Joan Verdera (Universitat Autònoma de Barcelona), 6 febbraio 2019, ore 10:00, aula 1BC45
Abstract: Dislocation Theory provides an energy on a planar unit mass distribution that contains a logarithmic term and an anisotropic term. Minimizers turn out to be unique (under certain assumptions) and compactly supported. The point of the talk is to explain how one can compute explicitely these minimizers in some cases. Singular integrals and the Plemelj jump formula play a role.
Spectral enclosures for block operator matrices, Francesco Ferraresso (Universität Bern, CH), 9 gennaio 2019, ore 14:30, aula 2BC30
Abstract: The spectrum of non self-adjoint unbounded operators is frequently studied in connection to the stability of the equilibria of evolutionary equations. It is known to be a difficult object to localize, already in the bounded case. The main problem is the lack of a good resolvent estimate in terms of the distance from the spectrum. It is then natural to restrict the class of operators in consideration to those presenting a specific structure, such as a block matrix decomposition. After recalling the relevant results from spectral theory, we will outline a general strategy to obtain enclosures for the spectrum of a block operator matrix via an abstract method based on the study of the so-called Schur complements. As an application, we will present a recent result about the linearized compressible Navier- Stokes system. The spectrum is enclosed inside a parabolic region lying in the left half-plane and decomposed in the union of the spectra of an operator with compact resolvent and of an holomorphic family of operators. This talk is based on an ongoing work with C. Tretter.
Spectral gaps for elastic and piezoelectric waveguides, Jari Taskinen (University of Helsinki), 22 ottobre 2018, ore 16:30, aula 1BC50
Abstract: We consider the band-gap structure of linear elasticity and piezoelectricity problems on quasiperiodic 3-dimensional waveguides. We consider waveguides with thin structures, which are created by thin ligaments connecting (infinitely many, translated copies of) bounded cells. We establish the existence of an arbitrary number of gaps, if the connecting ligaments of the cells are thin enough. In the case of the elasticity system we have quite precise information on the position of the spectral bands. The sharpest results are obtained using asymptotic analysis (work with F.Bakharev). In the case of the piezoelectricity system (work with S.Nazarov), the information is less precise, due to high complexity of the problem, and the mere existence of the band-gap structure for the essential spectrum needs a new proof, which we able to provide. Otherwise, the methods include a self-adjointreduction scheme, max-min-principle and weighted Sobolev estimates.
Boundary-Domain Integral Equations for Stokes PDE System in $L_p$-based spaces for Variable-Viscosity Compressible Fluid on Lipschitz Domain, Sergei Mikhailov (Brunel University London), 18 settembre 2018, ore 12:30, aula 1BC50
Abstract: In this presentation we consider Boundary-Domain Integral Equations (BDIEs) associated with the Dirichlet boundary value problem for the stationary Stokes system in $L_p$-based Sobolev spaces in a bounded Lipschitz domain in ${\mathbb R}^3$ with the variable viscosity coefficient. First, we introduce a parametrix and construct the corresponding parametrix-based variable-coefficient Stokes Newtonian and layer integral potential operators with densities and the viscosity coefficient in $L_p$-based Sobolev or Besov spaces. Then we generalize various properties of these potentials, known for the Stokes system with constant coefficients, to the case of the Stokes system with variable coefficients. Next, we show that the Dirichlet boundary value problem for the Stokes system with variable coefficients is equivalent to a BDIE system. Then we analyse the Fredholm properties of the BDIE systems in $L_p$-based Sobolev and Besov spaces and finally prove their invertibility in corresponding quotient spaces.
Strengthened extremal bounds for moduli of annuli under distortions by univalent and multivalent functions, Anatoly Golberg (Holon Institute of Technology), 17 settembre 2018, ore 12:00, aula 2AB40
Abstract: The theorems of TWB (Teichmueller-Wittich-Belinskiii) type imply the local conformality in the strong or weakened sense generating deep features of quasiconformal mappings at a prescribed point under assumptions of the finiteness of appropriate integral averages of the quantity $K_f(z) - 1;$ where $K_f(z)$ stands for the real dilatation coefficient. In the talk, we present the strengthened extremal bounds for distortions of the moduli of annuli in terms of integrals in TWB theorems under quasiconformal and quasiregular mappings and illustrate their sharpness by different examples. Several local conditions weaker than the conformality are also discussed.