ATMW Partial differential equations of fractional order (2015)

The workshop was conducted at TIFR Centre for applicable mathematics, Bangalore during  July 06­18, 2015. The response to our call for participation was very satisfying and we received seventy three applications and we could only select thirty of them, as planned in our original  proposal. The final number of outstation participants stood at twenty even and there was about  ten local participants as well. The programme was intended to introduce the theory of fractional order partial differential equations, an active area of research. The focus was to familiarise and  emphasise on the basics of the subject to the participants. 
The first week was primarily devoted to build the necessary background in aspects of analysis  which are essential for this subject. A number of important topics were discussed. The  fractional order Sobolev spaces were introduced and their main properties like compactness,  embeddings, density properties, etc. were taught in detail. In addition, in another series of  lectures, a comprehensive theory of pseudo differential operators were developed and it  included the aspects of tempered distributions and Fourier analysis. Moreover, aspects of  calculus of variations with connections to fractional order PDEs was discussed. 
Large part of second week was devoted to more specific problems involving fractional order  PDEs. The fractional Laplace operator was introduced and issues like their solvability,  maximum principle, etc., were discussed in detail. In addition, this subject has strong  connection with stochastic analysis and control theory. This particular connection was explored  in details in a separate series of lectures and particular emphasis was given on viscosity  solutions for such problems and its application on stochastic control problem . Another topic  covered was the uniqueness of positive solutions of a semilinear fractional order pde. One of  the recent and most difficult result proved in recent times was discussed.
All these courses were supported by tutorials covering exercises and and some of the  preliminary materials.

List of Speakers and topics covered :

1. Prof. Adimurthi , TIFR­CAM, Bangalore. ( AM)
   Various definitions of Fractional order Sobolev spaces, Sobolev inequalities, compactness, density properties, maximum principle, application to symmetry properties.
2. Dr. Imran Biswas, TIFR­CAM, Bangalore. ( IB)
   Stochastic analysis, control theory, connection with fractional order equations, viscosity  solutions.
3. Prof. KT Joseph, TIFR­CAM, Bangalore. (KTJ)
   Definition and basic properties of Pseudo differential operators, calculus of pseudo differential operators, asymptotic expansion of symbols, Schwartz kernel, oscillatory integrals, algebraic structures, continuity properties in Sobolev spaces, elliptic operators, Microlocal analysis.
4. Prof. K. Sandeep, TIFR­CAM, Bangalore. (KS)
   Semilinear fractional order pde, existence of solutions, symmetry of positive solutions, ground state solutions and their Morse index, uniqueness of ground state solutions.
5. Dr. Saugata Bandyopadhyay, IISER Kolkata. (SB)
   Various notions of convexities, classification of quasi convexity, minimizers for variational integrals, existence and uniqueness.
6. Dr. Venky Krishnan, TIFR­CAM, Bangalore. (VK)
   Definition and properties of distributions, tensor product of distributions, Fourier transform, Riesz potential, convolution, wavefront set,Radon Transform, Inverse Radon transform, application to image reconstruction problems.

Course Associates :

1. Arka Malik, TIFR­CAM, Bangalore. (AMa)
2. Debabrata Karmakar, TIFR­CAM, Bangalore. (DK)
3. Indranil Chowdury, TIFR­CAM, Bangalore. (IC)
4. Manmohan Vashisth , TIFR­CAM, Bangalore. (MV)

Time-Table

In addition to this list, around 10 local participants also attended the programme.

 How to reach TIFR, Bangalore