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Directorio
Espacio de trabajo digital
Laboratorios de investigación

Felix
ALI MEHMETI

  • Bâtiment ISTV 2
    Bureau B 0-4
My research activities are concerned with the following basic aspects:
 
1.       Analysis on multi-structures: network problems, transmission problems, interaction problems
2.       The transient tunnel effect
3.       Transient diffraction at semi-infinite cracks
4.       Nonlinear equations on singular domains
5.       Spectral theory
6.       Asymptotic Analysis
 
In this context the following novelties have been introduced
 
1.       An interaction concept with physical justification of coupling models
2.       A mathematical explication of the effects of retarded reflection and advanced transmission occurring in connection with the transient tunnel effect
3.       The limiting absorption principle for the Sommerfeld Problem, the solution of an open problem
4.       The theory of composition operators an function spaces on singular domains. Analysis of the influence of the boundary singularity on the convergence speed of iteration methods.
5.       The completeness of a family of generalized eigenfunctions of the Laplace operator on a star shaped network with semi-infinite rays
6.       Creation of a new variant of the stationary phase method with lossless error estimates. Creation of perturbation estimates for low and high frequencies for dispersive waves on certain semi-infinite networks. More generally the development of the principle to study the large time behavior of dispersive systems by frequency bands.
 
General aims of my research are the justification of models and the achievement of structural analytic information on physically motivated models composed of several possibly very different interacting media (basic aspect 1.). Examples are networks of transmission lines in micro-electronics and microwave theory, neural networks and interaction problems in quantum mechanics. One wants to take into account the properties of the media as well as those of the interaction zones. After developing a suitable model, one might be interested in existence and regularity of solutions, their explicit representations, time asymptotics, spatial asymptotics near the singularities and other qualitative results (basic aspect 6.).
 
For the modeling one might use the interaction concept, a functional analytic method which I developed for the description of the evolution of systems of interacting media.
 
Transmission and boundary value problems are special interaction problems. To investigate wave propagation in the vicinity of interfaces and boundaries one can cut off the rest and consider semi infinite geometries. This point of view with its conceptual advantages but also technical difficulties is a major issue of my research efforts (basic aspects 2. and 3.).
 
Interfaces and boundaries often are not smooth but polygonal or polyhedral for example. This aspect has been studied when analyzing the mapping properties of the composition operator on non-smooth domains and its applications on nonlinear elliptic and hyperbolic problems (basic aspect 4.).
 
Spectral theory of elliptic operators on semi-infinite geometries turns out to be technically difficult due to the appearance of generalized eigenfunctions. Satisfying results have been obtained in the cases of the semi-infinite crack (basic aspect 3.), the star-shaped network (basic aspect 5.) and the tadpole network. Typically, one has to examine the simultaneous influence of several parameters on oscillating integrals.
 
In applications, one meets cases where the time-space asymptotics of dispersive waves change nature abruptly depending on signal speed. In these cases usual methods of asymptotic expansions fail due to a lack of precision in the error estimates. For this reason we have concretised, modernized and refined the method of Erdélyi (aspect 6.). First applications concern wave functions of Schrödinger type with preferred initial speed. Other aspects are contained in perturbation results for small and large frequencies in the case of dispersive waves on star shaped and tadpole shaped networks (aspect (f)). In this type of results one combines solution formulas coming from spectral theory with asymptotic methods in frequency bands.

Fonctions actuelles

    In laboratory

  • Since 01/01/20201 :

    Responsable of the international relations

  • 01/04/2021 :

    Co-responsable du « Café Mathématique »

Diplômes universitaires

  • 1995 :
    Habilitation
  • 1987 :
    Doctorate

Expériences professionnelles

  • 1984 - 1988 :
    Assistant at the "Johannes Gutenberg-Universität Mainz", Germany
  • 1988 - 1990 :
    Research scholarship of the "German Research Association" (Deutsche Forschungsgemeinschaft, DFG)

  • 1990 - 1995 :
    Researcher at the ‘Technische Hochschule Darmstadt’, Germany, in a research team of the "German Research Association" (Deutsche Forschungsgemeinschaft, DFG)
  • 1996 :
    Guest professor at the University of Valenciennes
  • 1996 :
    Interim Professor (C4) at the University of Cologne
  • 1996 - :
    University Professor at the University of Valenciennes
  • 1998 - 2002 :
    Member of the scientific counsil of the University of Valenciennes
  • 2002 - 2006 :
    Member of the counsil of studies and life at the University of Valenciennes
  • 2006 - :
    Member of the counsil of the LAMAV
  • 2006 - 2014 :
    Adjoint director of the LAMAV
  • 2006 - 2008 :
    Member of the scientific counsil of the University of Valenciennes
  • 2008 - 2012 :
    Member of the administration counsil of the University of Valenciennes
  • 2008 - 2012 :
    Member of the commission of regulations and statutes of the University of Valenciennes
  • 2009 :
    Promotion to the grade of university professor, first class
  • 2015 - 2019 :
    Director of the LAMAV

Valorisations academiques

    Scientific responsibilities of theses

  • 1995-98 :
    Mihalincic, K. :
    “Time Decay Estimates for the Wave Equation With Transmission and Boundary Conditions”, defended october 30th, 1998 in Darmstadt, Germany
  • 1999-2002 :
    Régnier, V. :
    “Répartition du Flot d’Energie pour des Ondes Dispersives et Non-Dispersives sur des Espaces Ramifiés Elémentaires ou Localement Elémentaires de Dimension Un et Deux”, defended november 14th, 2002
  • 1998-2004 :
    Hagbe, J. F. :
    “Convergence speed of the Banach fixed point iteration for semilinear elliptic problems on conical domains”, defended june 17th, 2004
  • 1998-2004 :
    Daikh, Y. :
    “Temps de passage de paquets d’ondes de basses fréquences ou limités en bandes de fréquences par une barrière de potentiel”, defended september 16th, 2004.
  • 2013-16 :
    Dewez, F. :
    “Lossless estimates for asymptotic methods with applications to propagation features for dispersive equations”, soutenue le 3 novembre 2016, à l’Université Lille 1.

    • Belonging to learned societies

  • 1990 - :
    Deutscher Hochschulverband

1. Books

1.1. Research Monographs

[1] Ali Mehmeti, F. : Nonlinear Waves in Networks. Mathematical Research, vol. 80,
Akademie Verlag, Berlin 1994. 171 pages.
[2] Ali Mehmeti, F. : Transient Tunnel-Effect and Sommerfeld Problem ; Waves in
Semi-infinite Structures. Mathematical Research, vol. 91, Akademie Verlag, Berlin 1996.
210 pages.

1.2. Edited Books

[1] Ali Mehmeti, F., von Below, J., Nicaise, S. (eds.) : Partial Differential Equations
on Multistructures. Proceedings of the Conference at the CIRM in Luminy (Marseille,
France) 19.-24.5.1999, M. Dekker, lecture notes in pure and applied mathematics vol.
219, New York, Basel, 2001. 248 pages.

2. Articles in international journals

[1] Ali Mehmeti, F. : A characterization of a generalized C-infinity notion on nets ; Integral
Equations and Operator Theory 9 (1986), 753-766.
[2] Ali Mehmeti, F. : Regular Solutions of Transmission and Interaction Problems for
Wave Equations, Math. Meth. Appl. Sci., 11 (1989), 665-685.
[3] Ali Mehmeti, F. and S. Nicaise : Nonlinear Interaction Problems ; Nonlinear
Analysis, Theory, Methods and Applications 20, no.1, (1993), 27-61.
[4] Ali Mehmeti, F. : Spectral theory and L-infinity time decay estimates for Klein-Gordon
equations on two half axes with transmission : the tunnel effect ; Math. Meth. Appl. Sci.
17 (1994) no. 9, 697-752.
[5] Ali Mehmeti, F., Meister, E., Micalincic, K. : Spectral Theory for the Wave
Equation in Two Adjacent Wedges ; Math. Meth. Appl. Sci. 20 (1997) 1015-1044.
[6] Ali Mehmeti, F. and S. Nicaise : Nemytskij’s operators and global existence of small
solutions of semilinear evolution equations on nonsmooth domains ;
Comm. in P.D.E. 22(9& 10) (1997) 1559-1588.
[7] Ali Mehmeti, F., Nicaise, S. : Non-autonomous evolution equations on nonsmooth
domains ; Math. Nachr. 192 (1998) 37-70.
[8] Ali Mehmeti, F., Bochniak, M., Nicaise, S., Sändig, A.-M. : Quasilinear elliptic
systems of second order in domains with corners and edges : Nemytskij operator, local
existence and asymptotic behaviour ; Zeitschr. f. Anal. und ihre Anw. 21 (2002) 057-090.
[9] Ali Mehmeti, F., Régnier, V. : Splitting of energy of dispersive waves in a
star-shaped network ; Z. Angew. Math. Mech. 83 (2003) 2,105-118.
[10] Ali Mehmeti, F., Régnier, V. : Delayed Reflection of the Energy Flow at a Potential
Step for Dispersive Wave Packets ; Math. Meth. Appl. Sci. 27 (2004) 1145-1195.
[11] Ali Mehmeti, F., Régnier, V. : Global existence and causality for a transmission
problem with a repulsive nonlinearity ; Nonlinear Analysis 69 (2008) 408-424.
[12] Ali Mehmeti, F., Haller-Dintelmann, R., Régnier, V. : Multiple tunnel effect
for dispersive waves on a star-shaped network : an explicit formula for the spectral
representation ; J. Evol. Equ. 12, 2012, 513-545.
[13] F. Ali Mehmeti, K. Ammari, S. Nicaise, Dispersive effects and high frequency
behaviour for the Schrödinger equation in star-shaped networks ; Port. Math., 4 (2015),
309–355.
[14] F. Ali Mehmeti, K. Ammari and S. Nicaise, Dispersive effects for the Schrödinger
equation on the tadpole graph, J. Math. Anal. Appl., 448 (2017), 262–280.
[15] F. Ali Mehmeti, F. Dewez, Lossless error estimates for the stationary phase method
with applications to propagation features for the Schrödinger equation,
Math. Meth. App. Sci. 40 (3), 626-662 (2017).
[16] F. Ali Mehmeti, K. Ammari, S. Nicaise, Dispersive effects for the Schrödinger equation on finite metric graphs with infinite ends, J. Math. Phys. 65, 111503 (2024); doi: 10.1063/5.0183771
arXiv:2310.16628  (2024).

3. Comptes Rendus de l’Académie des Sciences de Paris

[1] Ali Mehmeti, F., Régnier, V. : Réflexion retardée pour des paquets d’ondes
dispersives sur un réseau en forme d’étoile ; C. R. Acad. Sci. Paris, Sér. I 337 (2003)
654-648.

4. Articles in proceedings with referee

[1] Ali Mehmeti, F. : Existence and Regularity of Solutions of Cauchy Problems for
Inhomogeneous Wave Equations with Interaction ; Operator Theory : Adv. and Appl.,
Vol. 50, Birkhäuser, Basel 1991, 23-34.
[2] Ali Mehmeti, F. and S. Nicaise : Some Realizations of Interaction Problems ; in :
Semigroup Theory and Evolution Equations, P. Clément, E. Mitidieri, B. de Pagter
(eds.), L.N. in Pure and Appl. Math. 135, M. Dekker, New York, Basel, Hong Kong 1991,
15-28.
[3] Ali Mehmeti, F. and S. Nicaise : Characterization of iterated powers of operators in
nonsmooth domains and Nemetskij’s operators ; in : G. Lumer, S. Nicaise, B.-W. Schulze
(eds.), Partial Differential Equations ; Models in Physics and Biology ; Mathematical
Research, vol. 82 ; Akademie Verlag, Berlin 1994.
[4] Ali Mehmeti, F. and S. Nicaise : Banach algebras of functions on nonsmooth
domains ; Operator Theory : Advances and Applications, Vol. 102 (1998) 11-20.
[5] Ali Mehmeti, F., Dekoninck, B. : Transient vibrations of planar networks of
beams : interaction of flexion, transversal and longitudinal waves ; in : ‘Partial
Differential Equations on Multistructures’, M. Dekker, lecture notes in pure and applied
mathematics vol. 219, p. 1-18, New York, Basel, 2001.
[6] Ali Mehmeti, F., Haller-Dintelmann, R., Régnier, V. : Expansions in
generalized eigenfunctions of the weighted Laplacian on star-shaped networks ; in :
H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, J. von Below (eds.) :
Functional Analysis and Evolution Equations. The Günter Lumer Volume. 1-16,
Birkhäuser, Basel, 2008.
[7] Ali Mehmeti, F., Haller-Dintelmann, R., Régnier, V. : The influence of the
tunnel effect on L-infinity time decay. W. Arendt, J. A. Ball, J. Behrndt, K.-H. Förster,
V. Mehrmann, C. Trunk (eds) : Spectral Theory, Mathematical System Theory,
Evolution Equations, Differential and Difference Equations : IWOTA10 ; Springer, Basel ;
Operator Theory : Advances and Applications, 221, 2012, 11-24.
[8] Ali Mehmeti, F., Haller-Dintelmann, R., Régnier, V. : Energy flow above the
threshold of tunnel effect. A. Almeida, L. Castro, F.-O. Speck (eds.) : Advances in
Harmonic Analysis and Operator Theory : the Stefan Samko Anniversary Volume ;
Springer, Basel ; Operator Theory : Advances and Applications, 229, 2013, 65-76.
[9] Ali Mehmeti, F., Haller-Dintelmann, R., Régnier, V. : Dispersive waves with
multiple tunnel effect on a star shaped network. Discrete and Continuous Dynamical
Systems, Series S, 6,3, 2013, 783-791.

5. Articles in proceedings

[1] Ali Mehmeti, F. : Problèmes de transmission pour des équations des ondes linéaires et
quasilinéaires ; Séminaire Equations aux Dérivées Partielles Hyperboliques et
Holomorphes 1982/83, J. Vaillant (ed.), Travaux en Cours, Hermann, Paris 1984, 75-96.
[2] Ali Mehmeti, F. : Linear and Nonlinear Transmission and Interaction Problems ;
Semesterbericht Funktionalanalysis, Band 13, Wintersemester 1987/88, G. Greiner, R.
Nagel, H.H. Schaefer, U. Schlotterbeck, M. Wolff (eds.), Eberhard-Karls-Universität
Tübingen 1988, 119-134.
[3] Ali Mehmeti, F. : Global Existence of Solutions of Semilinear Evolution Equations
with Interaction ; Symposium ‘Partial Differential Equations’, Holzhau 1988, B.-W.
Schulze, H. Triebel (eds.), Teubner-Texte zur Mathematik 112, BSB B.G. Teubner,
Leipzig 1989, 11-23.
[4] Ali Mehmeti, F. : Propagation of singularities for an interaction problem ;
Vorlesungsreihe, Workshop on Nonlinear Hyperbolic Problems (R. Leis, R. Racke, eds.)
Abstracts No. 15, Bonn 1991, 8-10.
[5] Ali Mehmeti, F. and S. Nicaise : Compact Imbeddings and Interaction Problems ;
Semesterbericht Funktionalanalysis, Workshop on Operator Semigroups and Evolution
Equations, Blaubeuren, 1989, G. Greiner, R. Nagel, F. Räbiger, U. Schlotterbeck (eds.),
Eberhard-Karls-Universität Tübingen 1991, 143-152.
[6] Ali Mehmeti, F. : Reflection and refraction of singularities for wave equations with
interface conditions given by Fourier integral operators ; Symposium ‘Analysis on
Manifolds with Singularities’, Breitenbrunn 1990, B.-W. Schulze, H. Triebel (eds.),
Teubner-Texte zur Mathematik 131, BSB B.G. Teubner, Leipzig (1992) 6-19.
[7] Ali Mehmeti, F. : Existence and regularity of solutions of quasilinear wave equations
on one-dimensional networks ; in : U. Helmke, R. Mennicken, J. Saurer (eds.), Systems
and Networks : Mathematical Theory and Applications, Vol. II, Mathematical Research,
Vol. 79, Akademie Verlag, Berlin 1994, 791-792.

6. Theses

[1] Ali Mehmeti, F. : Transmissionsprobleme für lineare und quasilineare
Wellengleichungen ; diploma thesis, Johannes Gutenberg-Universität Mainz 1982.
[2] Ali Mehmeti, F. : Lokale und globale Lösungen linearer und nichtlinearer
hyperbolischer Evolutionsgleichungen mit Transmission ; doctoral thesis, Johannes
Gutenberg-Universität Mainz 1987.
[3] Ali Mehmeti, F. : Transient Waves in Semi-infinite Structures : the Tunnel-Effect and
the Sommerfeld Problem ; Technische Hochschule Darmstadt, habilitation thesis ;
Preprint-Nr. 1762, Darmstadt, Juli 1995.

President of the Music School in Artres "Société l'Avenir"