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Mathematical Modeling and Analysis

The research group is composed of active researchers with diverse backgrounds in mathematical analysis and mathematical modelling with applications to biology and ecology. Our three main domains of interest are mathematical ecology, complex analysis and partial differential equations.

Mathematical Ecology

Mathematical ecology is an interdisciplinary research area where advanced mathematical and computational tools are used to study issues in ecology and evolution. At Umeå, research on these topics involved not only members of the Analysis and Mathematical Modelling group from the department of Mathematics, but also researchers from the Department of Ecology and Environmental Science (EMG) at the interdisciplinary Icelab

Partial Differential Equations

Our research on partial differential equations (PDE) are mainly focused on estimating p-harmonic functions and proving existence and uniqueness theorems for fully nonlinear PDEs. The p-harmonic functions are solutions to the p-Laplace equation and are natural nonlinear generalizations of the well known harmonic functions. p-harmonic functions have connections to e.g., minimization problems, nonlinear elasticity theory, fluid dynamics, stochastic games and image processing. A typical research problem we consider is; how to determine growth rates of positive p-harmonic functions which vanish on a portion of the boundary of some domain?

Using the theory of viscosity solutions we work on proving existence and uniqueness theorems for fully nonlinear PDEs. The setting of the problems are usually general, including e.g. parabolic p-Laplace-type equations in time-dependent domains and systems of PDEs with interconnected obstacles. The latter has applications within optimal switching problems which are nowadays frequently studied in mathematical finance. We also conduct research on existence and uniqueness of solutions to stochastic differential equations, which, for example, are frequently used when modeling particle dispersion in the atmosphere.

Besides creative elaborate constructions, our research on differential equations involves real analysis, integration theory, measure theory, regularity theory, probability theory etc.

Complex Analysis

We do research mainly in several complex variables and our research is focused upon pluripotential theory and certain aspects of analytic function theory. Pluripotential theory is a nonlinear analog of the classical potential theory for subharmonic functions. The main objects of study are plurisubharmonic functions and the complex Monge-Ampere operator (ddcu)n – a generalization of Laplace's operator. In the theory of analytic functions we are mainly interested in ideals and spectrum for algebras of analytic functions, in particular Gleason's problem. In connection with this, we are also interested in domains of existence and envelopes of holomorphy for classes of analytic functions.