Projects / Programmes
Zvezni in diskretni sistemi v nelinearni analizi (Slovene)
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
nonlinear partial differential equation, differential operator, variational analysis, qualitative analysis of solutions
Researchers (13)
Organisations (2)
Abstract
The proposed 3-year research project aims at strengthening the connections between more fundamentally oriented areas of mathematics like topology, functional analysis, calculus of variations, and nonlinear analysis, and the more applied oriented and more recently emerging disciplines of discrete mathematics, mathematical physics, and optimization.
The overall goal of the project is to obtain, with methods from fundamental mathematics, new effective tools to unravel the complexity of some nonlinear phenomena arising in applied sciences. We are mainly concerned to study the dualism between "continuous" and "discrete" models and we are interested in the mathematical analysis of two classes of nonlinear problems: (i) partial differential equations described by continuous differential operators (homogeneous or non-homogeneous); (ii) partial difference equations describing discrete systems.
Our study is motivated by many complex phenomena that are described by nonlinear partial differential or difference equations, the solutions of which exhibit nonstandard behavior like concentration, oscillation, blow-up. Singular or degenerate phenomena and the behavior of systems at or near criticality will play a central place in our analysis.
The research will address fundamental issues like existence and stability of solutions, optimal regularity, perturbation effects, asymptotics but will mainly be devoted to the qualitative and quantitative analysis of multiscale phenomena in stationary and evolution systems. We shall focus on nonlinear systems with geometric structures, where the dynamics is driven by energy functionals.
We are concerned with some nonstandard models like systems dealing with irregular geometric structures (fractals), anisotropic problems described by nonhomogeneous differential or difference operators, singular or degenerate phenomena.
The novel features of our approach are the use of topological, variational and analytic methods combined with a deep geometric approach of the structures involved in the models described by various classes of nonlinear systems. The proofs combine several refined tools including critical point theory, Morse theory, deformation methods, index theory, Ljusternik-Schnirelmann theory, category, monotonicity methods, Karamata regular variation theory, numerical methods.
The proposal will open whole fields of investigation in nonlinear partial differential equations in the future, clarify and simplify our knowledge on the behavior of solutions of these problems and provide applied scientists some new insight on these models. Relevant applications include nonlinear models in non-Newtonian 'smart' fluids, image restoration, robotics (systems with variable exponent), chaos theory, earthquakes, Brownian motion, image processing (PDEs on fractals), Riemannian geometry, mathematical biology (blow-up and singular phenomena), biological neural networks, heat diffusion, temperature distribution, cellular neural networks (difference equations).
We shall continue to publish our results in top journals and write monographs on nonlinear analysis and its applications with leading publishers. Our investigations will be done in intensive collaboration with leading research groups from the European Union, the United States, Russian Federation, China and Japan, in the framework of international projects and networks.
We shall continue to organize conferences and workshops in Slovenia and abroad, with participation of several leading foreign experts, at which new results will be presented and an intensive exchange of expertise will be enabled.
We shall continue our excellent collaboration with business users - small and big companies in Slovenia and abroad. We shall further develop the PhD program in nonlinear analysis and applications in Slovenia and shall intensively include PhD students in our research. We shall also continue to devote a significant amount of time to the popularization of science in Slove
Significance for science
The research area of our project group belongs to one of the fundamental fields of mathematics and the results of our research will be significant contributions to the worldwide mathematical knowledge. We shall employ the latest developments and the relevance of our work will be demonstrated by many citations. Several of our results will be applicable also in other areas of mathematics as well as in other areas of science.
A very important purpose of our research project is to continue the significant development of the excellent competitive research in nonlinear analysis in Slovenia, in particular partial differential equations and the calculus of variations, to reach the level of the best similar centers in Europe and elsewhere and participate in the world research network, especially in the European Union.
The proposed project will also substantially advance our common knowledge in the field of nonlinear analysis, both pure and applied. As it follows from the name of this project, it is the field where the modern field of nonlinear analysis helps to analyze in a rigorous way concrete models proposed by the applied sciences. Nonlinear analysis is one of the areas of basic research having most potential for broad affirmation in the international scientific society. In recent years, many renowned scientists have successfully discovered new ways to apply the abstract results in this field. Note that both 2015 Abel Prize Laureates, John F. Nash and Louis Nirenberg, received their prizes „for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications”.
Members of this research team have already discovered new applications of a branch of nonlinear analysis, namely the Karamata regular variation theory applied for the first time to the precise asymptotic analysis of solutions of some important classes of nonlinear elliptic equations. Recall that the Karamata theory was initially introduced in the 1930s due to its applications to probability theory. Our original results on problems with variable exponent or nonlocal problems have great potential for applications in the study of several concrete phenomena.
The proposed research problems have been in the center of attention by many leading experts in nonlinear analysis around the world in the last few years. We are proposing new methods and techniques for resolving very difficult unsolved problems. The outstanding references of the award winning senior members of our research team guarantee the success of the proposed research. In items 8 and 9 of this application we list our most important achievements and awards, the great impact of our results and the worldwide acknowledgement of their importance by the scientific community.
Significance for the country
Our project group has had a long and successful history of cooperation with business since the 1980's. We mention here that the project leader was elected to the Slovenian Engineering Academy (IAS) for his outstanding long-term cooperation with the Slovenian industry. Members of our group have so far collaborated on the following applied projects:
(i) Telecommunications: with the telephone company Iskra Telematika research laboratories on optimization of performance of digital telephone switchboards.
(ii) Electrotechnics: with the company Elektrina on randomness verification and irregularity analysis of a mechanical random generator.
(iii) Optimization of shapes: with the ship building company Seaway group on optimization methods for shapes of yacht hulls.
(iv) Process optimization: with companies Abelium and Epilog on devising new technological solutions in optimization of the logistics of large storage capacities.
(v) Pharmaceutical industry: with the main national pharmaceutical company Krka on various statistical analyses.
(vi) Electronics: with company Elatec on algorithms for digital scanning probe microscopes controller with feedback for regulation of vertical moves.
(vii) Optimization and logistics: with high-tech company Abelium and start-up company Epilog on integrated software solutions for internal logistics and automated warehouses.
(viii)Fraud and anomalities detection: with company Optilab on technological solutions in the field of fraud and anomalies detection with efficient support.
(ix) Textile industry: with the Faculty of Natural Sciences and Engineering, University of Ljubljana, on analyzing the influence of chemical finishes on the surface properties of glass plate.
We shall continue our strong business collaboration with industry, applying our outstanding expertise in nonlinear analysis and its applications to various problems of big and small companies, and other business users, in Slovenia and abroad.
Most important scientific results
Interim report,
final report
Most important socioeconomically and culturally relevant results
Interim report,
final report
