كلية العلوم الدقيقة والتطبيقية
مخبر الرياضيات الاساسية والتطبيقية لوهران
مدير المخبر: Abderrahmane SENOUSSAOUI
الوصف
The laboratory is organized into six (6) closely collaborating research teams:
- Team for Operator Analysis, Resonance, and Fredholm Theory.
- Spectral Analysis of Partial Differential Equations.
- Control Theory of Partial Differential Equations and Optimization.
- Nonlinear Boundary Value Problems.
- Qualitative Study of Partial Differential Equations and History of Conics.
- Analysis, Geometry, and Applications.
Each team consists of experienced faculty researchers and doctoral students pursuing traditional PhDs and/or LMD degrees. Fifty-six researchers are affiliated with the laboratory, including twelve (12) Professors, sixteen (16) Class A Associate Professors, eleven (11) Class B Associate Professors, and seventeen doctoral students.
The laboratory conducts research in various areas and fields of fundamental
and applied mathematics: operator theory, spectral theory, partial differential equations, ordinary and abstract differential equations, essential spectra and Fredholm theory, microlocal analysis, nonlinear PDEs, resonance theory, nonlinear analysis and boundary value problems, PDE control, and optimization with concrete applications to optimal control problems and those in mathematical physics.
For more details, given a generally unbounded operator on a Hilbert space, we define—in addition to the list of various essential spectra known in the mathematical literature—the semiregular essential spectrum and the generalized Kato essential spectrum,.... We are interested in the topological and qualitative aspects of these new spectra, and we compare them with the other essential spectra. We quantify this comparison using the set-theoretic symmetric difference.
One of the central questions is the invariance of the various essential spectra under (additive) perturbations. Other relevant and quite interesting questions involve addressing the matrix case and testing the “Spectral Mapping Theorem” on this collection of essential spectra; it also involves examining these spectra for the sum and product of operators.
Regarding this last point, it is known that linear evolution operators (abstract Cauchy problems) defined on a product space involve matrix elements that can be decomposed into sums and products of closed operators; however, they are generally neither closed nor closable. This instability in closability poses a major obstacle in many problems in spectral analysis and PDEs. To address this, we introduce a new class of linear operators, called quotient operators of bounded operators. Quotient operators are unbounded operators on which we impose a topological condition inspired by the concept of the graph of a linear operator. In particular, we show that this class is stable with respect to standard operations: finite and infinite sums, products, and limits. We primarily aim to study the properties of quotient operators, such as: closure and closability, adjoint, symmetry, essentially self-adjoint, self-adjoint, various types of normality, Fredholm property, invertibility, spectrum and resolvent, as well as functional calculus, etc. It also seems very interesting to address the question of the quotient of unbounded operators by essentially recovering the topological characterization of this new class of operators through the study of operator extensions.
Furthermore, the essential tools for introducing resonance theory are Fredholm theory via the essential spectrum, microlocal analysis (pseudodifferential operators and Fourier integrals), and the expression of Hamiltonians in Birkhoff normal form.
Within the scope of this project, we also aim to:
- Analyze the class of pseudodifferential operators and that of semiclassical Fourier integral operators. Developments of Schrödinger operators in Birkhoff normal form will be carried out; in particular, we will express the different types of resonances near an equilibrium position by explicitly and originally calculating the coefficients of the Birkhoff normal form on Schrödinger Hamiltonians.
- Study the existence of solutions to nonlinear boundary value problems governed by ordinary or partial differential equations. We will attempt to establish necessary and sufficient conditions guaranteeing the existence of solutions to this type of problem, with a particular focus on positive periodic solutions. We will also address the numerical solution of these nonlinear problems.
- Identify various related control problems, such as identifiability issues and the study of approximate control for the semilinear heat equation with and without control constraints; boundary controllability of distributed systems with mixed and incomplete data; construction of boundary sentinels for systems with mixed data; construction of discriminant sentinels; study of no-regret control for the heat equation with delay.
- Characterize the weak lower Lyapunov semicontinuity for functions associated with first-order differential inclusions in Hilbert spaces. We are also interested in the study of finite and infinite discrete-time deterministic models in optimization within a non-smooth framework, and we establish approximate optimality conditions for deterministic models based on subdifferential calculus in various situations.
The expected results have applications in several areas of pure and applied mathematics, including spectral theory, partial differential equations, ordinary and abstract differential equations, quantum mechanics (mathematical physics), semiclassical analysis, and optimal control. Typical applications will be explored in transport equations, Schrödinger equations, wave equations, and state equations in resonance theory, as well as in the control and optimization of nonlinear problems.
- The team's scientific and educational work centers on the theme “Qualitative Analysis of Partial Differential Equations and the History of Conic Sections.” Four main areas form the core of this research program: nonlinear hyperbolic and elliptic partial differential equations, conservation laws, the Born-Oppenheimer approximation and pseudodifferential calculus, as well as the history of mathematics. Among the topics addressed are the existence and multiplicity of solutions for a certain class of nonlinear elliptic equations using variational methods, Ekeland’s principle and the neck theorem, the decay of expansion waves, and the construction of retrograde characteristics for solutions to certain Riemann problems, as well as the study of the geometry of triangles and conics by mathematicians of the Arab-Muslim civilization.
- Nonlinear PDEs with singular Hardy-type weights and critical Sobolev exponents; variational methods are the necessary tools for this theory; the goal is to express the solution to a boundary value problem as the mathematical expectation of a random quantity linked to a Markov chain.
- Characterization of curves and surfaces in Riemannian and Lorentzian manifolds, particularly those of dimension 3 and higher dimensions. More specifically, it seeks to naturally associate their coordinate systems with the corresponding curvature functions for curves and with developability and singularities for surfaces. The second focuses on properties and structures that require a differentiable structure on a manifold, such as the Ricci tensor, Ricci solitons, and applications to modern physics.
- Study of symmetries in various geometric spaces, which is one of the most interesting topics in geometry and mathematical physics.
- Team for Operator Analysis, Resonance, and Fredholm Theory.
- Spectral Analysis of Partial Differential Equations.
- Control Theory of Partial Differential Equations and Optimization.
- Nonlinear Boundary Value Problems.
- Qualitative Study of Partial Differential Equations and History of Conics.
- Analysis, Geometry, and Applications.
Each team consists of experienced faculty researchers and doctoral students pursuing traditional PhDs and/or LMD degrees. Fifty-six researchers are affiliated with the laboratory, including twelve (12) Professors, sixteen (16) Class A Associate Professors, eleven (11) Class B Associate Professors, and seventeen doctoral students.
The laboratory conducts research in various areas and fields of fundamental
and applied mathematics: operator theory, spectral theory, partial differential equations, ordinary and abstract differential equations, essential spectra and Fredholm theory, microlocal analysis, nonlinear PDEs, resonance theory, nonlinear analysis and boundary value problems, PDE control, and optimization with concrete applications to optimal control problems and those in mathematical physics.
For more details, given a generally unbounded operator on a Hilbert space, we define—in addition to the list of various essential spectra known in the mathematical literature—the semiregular essential spectrum and the generalized Kato essential spectrum,.... We are interested in the topological and qualitative aspects of these new spectra, and we compare them with the other essential spectra. We quantify this comparison using the set-theoretic symmetric difference.
One of the central questions is the invariance of the various essential spectra under (additive) perturbations. Other relevant and quite interesting questions involve addressing the matrix case and testing the “Spectral Mapping Theorem” on this collection of essential spectra; it also involves examining these spectra for the sum and product of operators.
Regarding this last point, it is known that linear evolution operators (abstract Cauchy problems) defined on a product space involve matrix elements that can be decomposed into sums and products of closed operators; however, they are generally neither closed nor closable. This instability in closability poses a major obstacle in many problems in spectral analysis and PDEs. To address this, we introduce a new class of linear operators, called quotient operators of bounded operators. Quotient operators are unbounded operators on which we impose a topological condition inspired by the concept of the graph of a linear operator. In particular, we show that this class is stable with respect to standard operations: finite and infinite sums, products, and limits. We primarily aim to study the properties of quotient operators, such as: closure and closability, adjoint, symmetry, essentially self-adjoint, self-adjoint, various types of normality, Fredholm property, invertibility, spectrum and resolvent, as well as functional calculus, etc. It also seems very interesting to address the question of the quotient of unbounded operators by essentially recovering the topological characterization of this new class of operators through the study of operator extensions.
Furthermore, the essential tools for introducing resonance theory are Fredholm theory via the essential spectrum, microlocal analysis (pseudodifferential operators and Fourier integrals), and the expression of Hamiltonians in Birkhoff normal form.
Within the scope of this project, we also aim to:
- Analyze the class of pseudodifferential operators and that of semiclassical Fourier integral operators. Developments of Schrödinger operators in Birkhoff normal form will be carried out; in particular, we will express the different types of resonances near an equilibrium position by explicitly and originally calculating the coefficients of the Birkhoff normal form on Schrödinger Hamiltonians.
- Study the existence of solutions to nonlinear boundary value problems governed by ordinary or partial differential equations. We will attempt to establish necessary and sufficient conditions guaranteeing the existence of solutions to this type of problem, with a particular focus on positive periodic solutions. We will also address the numerical solution of these nonlinear problems.
- Identify various related control problems, such as identifiability issues and the study of approximate control for the semilinear heat equation with and without control constraints; boundary controllability of distributed systems with mixed and incomplete data; construction of boundary sentinels for systems with mixed data; construction of discriminant sentinels; study of no-regret control for the heat equation with delay.
- Characterize the weak lower Lyapunov semicontinuity for functions associated with first-order differential inclusions in Hilbert spaces. We are also interested in the study of finite and infinite discrete-time deterministic models in optimization within a non-smooth framework, and we establish approximate optimality conditions for deterministic models based on subdifferential calculus in various situations.
The expected results have applications in several areas of pure and applied mathematics, including spectral theory, partial differential equations, ordinary and abstract differential equations, quantum mechanics (mathematical physics), semiclassical analysis, and optimal control. Typical applications will be explored in transport equations, Schrödinger equations, wave equations, and state equations in resonance theory, as well as in the control and optimization of nonlinear problems.
- The team's scientific and educational work centers on the theme “Qualitative Analysis of Partial Differential Equations and the History of Conic Sections.” Four main areas form the core of this research program: nonlinear hyperbolic and elliptic partial differential equations, conservation laws, the Born-Oppenheimer approximation and pseudodifferential calculus, as well as the history of mathematics. Among the topics addressed are the existence and multiplicity of solutions for a certain class of nonlinear elliptic equations using variational methods, Ekeland’s principle and the neck theorem, the decay of expansion waves, and the construction of retrograde characteristics for solutions to certain Riemann problems, as well as the study of the geometry of triangles and conics by mathematicians of the Arab-Muslim civilization.
- Nonlinear PDEs with singular Hardy-type weights and critical Sobolev exponents; variational methods are the necessary tools for this theory; the goal is to express the solution to a boundary value problem as the mathematical expectation of a random quantity linked to a Markov chain.
- Characterization of curves and surfaces in Riemannian and Lorentzian manifolds, particularly those of dimension 3 and higher dimensions. More specifically, it seeks to naturally associate their coordinate systems with the corresponding curvature functions for curves and with developability and singularities for surfaces. The second focuses on properties and structures that require a differentiable structure on a manifold, such as the Ricci tensor, Ricci solitons, and applications to modern physics.
- Study of symmetries in various geometric spaces, which is one of the most interesting topics in geometry and mathematical physics.