Projects / Programmes
Selected problems of nonlinear analysis
| 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; function spaces with variable
exponent; nonhomogeneous differential operator; variational analysis
Researchers (13)
Organisations (2)
Abstract
This research project is at the interplay between pure and applied nonlinear analysis. It focuses on the analytical study of nonlinear partial differential equations describing models arising in the applied sciences. It concentrates on several specific classes of nonlinear systems, with the aim of providing physically meaningful results by using rigorous analytical tools.
In this research project, we are concerned with the study of two classes of nonlinear partial differential equations (PDE's): (i) elliptic and evolutionary problems with variable exponent and (ii) nonlocal fractional equations. In both cases, we are interested in the qualitative, asymptotic and bifurcation analysis of solutions. A particular interest will be given to the understanding of new phenomena involving these nonlinear problems.
In the first case, we are interested in several competition and perturbation effects for nonlinear equations with one or more variable exponents. We take into account the spectral characterization of nonhomogeneous differential operators, Sobolev-type critical exponents, indefinite potentials with possible singularities, and bifurcation analysis of solutions. A new phenomenon in this kind of new problems concerns various concentration phenomena of the spectrum, namely the existence of a continuous spectrum concentrating either near the origin or in a neighbourhood of infinity. The study of nonlinear problems with variable exponents is motivated by relevant applications to non-Newtonian electro-rheological (smart) fluids, image restoration, robotics, or in related models arising in mathematical physics and other applied fields.
Nonlocal problems involving the fractional Laplace operator arise in the description of various phenomena, such as plasma physics, flame propagation, Hamilton-Jacobi with critical fractional diffusion, or phase transitions in the Gamma convergence framework. From a probabilistic point of view, the fractional Laplace operator is the infinitesimal generator of a Lévy process. We are interested in the qualitative analysis of nonlocal Schrödinger and Kirchhoff-type equations. Our main purpose is to continue and to extend our numerous contributions to this field and to develop a rigorous variational and topological analysis of solutions. We take into account the following directions: a better understanding of critical phenomena of Brezis-Nirenberg type; study of the supercritical case; combined effects in nonlocal fractional equations; bifurcation analysis of solutions.
Our analysis combines refined tools from nonlinear analysis, including variational methods (critical point theory, Morse theory, energy estimates), topological methods (Ljusternik-Schnirelmann theory, deformation methods, category and genus), monotonicity methods (nonlinear maximum principle, comparison principles), or asymptotic analysis (Karamata regular variation theory). In all cases, we are interested in finding the most natural hypotheses and to study only models arising in the applied sciences.
We shall publish our results in excellent journals of pure and applied mathematics and publish several new monographs on nonlinear analysis and its applications for leading scientific 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 the current (and future) international (bilateral and multilateral) research projects and networks. We are planning a conference and a summer workshop in Slovenia, with participation of several leading foreign experts, at which new results will be presented and an intensive exchange of expertise will be enabled. We plan various applications of our new results outside mathematics. We shall further develop the PhD program in nonlinear analysis and applications in Slovenia and shall intensively include PhD students in our research.
Significance for science
The central purpose of this research project is to contribute to the development of a very competitive research school in nonlinear analysis in Slovenia, at the level of the best similar centers in Europe and in the world. We intend to have a very positive influence on further intense development of the Slovenian mathematical school, emphasizing partial differential equations, the calculus of variations, and their bonds to the world research network, especially in the European Union.
The proposed project will 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. Let us recall that both Abel Prize Laureates 2015, 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”.
Some of the members of this research team 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. Let us recall that the Karamata theory has been 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. The references of the members of our research team guarantee the success of the proposed research. We are proposing new methods and techniques for resolving very difficult unsolved problems.
Our research group is well established in its area and has received many domestic and foreign awards. We expect that our results will continue to be published in leading specialized journals and that the leading members of our group will continue to receive invitations to give lectures at important international conferences, confirming the international recognition of our research group.
We expect to have increased interest of eminent foreign research institution in cooperation with our institute, especially from European Union. At present our research group has the largest number of international projects among all Slovenian research groups in the area of mathematics.
Significance for the country
We have been, for several years, intensively cooperating with several applied sectors. For example, the mathematical model associated to an asymmetric encryption system that we developed, has been successfully applied to the data transfer between communication systems in order to guarantee the security of information systems. In these areas we closely cooperate with industry in order to create an experimental model of a portal that implements authentication functions based on proprietary mathematical models superior to the standard ones. Therefore we expect such collaboration also in the future.
The research proposed in this project, will have extraordinary positive effect on the development of graduate studies in Slovenia, at every university with the PhD program in mathematics. This is especially true for education of future collaborators of our research group at the Universities of Ljubljana, Maribor, Nova Gorica and Primorska. Under the mentorship of our researchers and distinguished foreign researchers, our young researchers will prepare their PhD's on the most up-to-date topics of modern nonlinear analysis.
The research area of this project group belongs to one of the fundamental fields of mathematics and the results of our research will be contributions to the worldwide mathematical community; they will include the latest developments and their relevance will be demonstrated by many citations. Some of the results of the group will also be applicable in other areas of mathematics.
All this research will be related to and building upon our past successful research in this field and it will be connected with the problems which have been very successfully studied with very positive feedback in numerous international projects of our research team. As the result of our longstanding efforts our institute is an internationally renowned European center of pure and mixed pure and applied research, at the interplay between nonlinear analysis, topology and geometry.
Most important scientific results
Interim report,
final report
Most important socioeconomically and culturally relevant results
Interim report,
final report
