The theory of viscosity solutions is one of the most central and fastest growing areas in the field of nonlinear partial differential equations.  Its scope extends from rather deep and challenging mathematical problems to the modelling and analysis of a large number of applied problems in science, engineering, atmospheric/ocean studies and economics, which include, among others, phase transitions, moving fronts, statistical mechanics, combustion, turbulent reaction-diffusions, optimal and stochastic control, mathematical finance, image processing, etc.  The understanding of such applications obviously requires, in addition to theoretical results, extensive computations and efficient numerical methods, which can be derived and analysed using the theory of viscosity solutions. 

The long list of applied areas and problems which can be studied using methods related to viscosity solutions, the rather difficult and challenging mathematical problems that encompasses as well as the astonishing progress already achieved, in such a relatively short (for mathematics) period, using viscosity solutions has generated an ever growing interest in Europe, US, Japan and China.  An indication of the growing importance of this theory for mathematics is that, since 1981 
when M. G. Crandall and P.-L. Lions introduced the concept of the viscosity solution, more than one thousand papers related to the subject have been published in mathematical journals of high quality, a rather amazing fact for mathematics.  A proof of the importance and relevance of viscosity solutions for applied mathematics is the fact that they have already been used successfully to analyse a large number of questions related to the applications mentioned above. 

Europe is in a unique position to be at the forefront of this theory.  Some of the leading groups in the world in the theory of viscosity solutions and its applications are located in Europe and all of them are participating in this project.  Bringing them together increases the talent, the ability and the expertise needed to address the new problems.  The resulting interaction among groups working in some aspects of the applications will definitely increase the understanding and provide the necessary input and ability to address the even more challenging problems lying ahead of us.  Besides its mathematical importance, further progress in the theory of viscosity solutions and its applications will lead to better, and, in some occasions, important, understanding in sciences, medecine, engineering and in socio-economic issues. 

Finally carrying out research in the theory of viscosity solutions and the activities surrounding this research will result in educating a whole new generation of young researchers, the so-called New Applied Mathematicians, who will be able to do modelling, theoretical work at the highest possible level of Mathematics and numerical computations, and will be involved in real problems.  
 
Partnership  - 10 working teams from 6 countries: 
 

Last modified: Wednesday, 30-Sep-1998 13:31:05 CEST