Inverse problems in Mathematical Biology, Chemistry and Technology

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Our results may be summarized as follows:1. First, we studied the functional analytic approach to the problem of construction of Markov processes with Ventcel' boundary conditions in probability theory. Second-order elliptic differential operators with discontinuous coefficients enter naturally in connection with the problem of construction of Markov processes in probability. Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of Calderon and Zygmund of singular integral operators with non-smooth kernels. The presentation of these new results is the main purpose of our research.2. Secondly, we studied existence and uniqueness problems of positive solutions of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We discussed the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and proved that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.