Monotonicity formula methods for nonlinear PDEs

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It is known that for linear elliptic and parabolic equations of second order one can construct monotone functions from the solution. A typical example is the mean value integral of harmonic function over a ball. In this case the mean value integral is monotone function of the radius of the ball. There are more complex examples of this sort such as Almgren's frequency formula which, among other things, helps to identify the structure of the zero set of harmonic function. The aim of this project is to construct monotone functions for the solutions of some nonlinear equations. The choice of this type of operators is adequate since there are various physical problems where the nonlinear equations emerge. For instance, the flow of non-Newtonian fluids with power law dependence of the shear tensor from the velocity, the flow of gas in porous media in turbulent regime, the quantum field theory and the interaction of two biological groups without self-limiting. We aim to construct monotone functions for three free boundary problems with nonlinear governing equations and point out some applications in stochastic game theory (Tug-of-War model), Chemical Kinetics and Combustion (smouldering of cigarettes and flame propagation).