Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

Strict Lyapunov functions for semilinear parabolic partial differential equations

For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a period...

Symmetry Coefficients of Semilinear Partial Differential Equations

We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m > 1 and the equation is also linear in its derivatives of order m− 1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial...

Boundary element monotone iteration scheme for semilinear elliptic partial differential equations

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration schem...

On Markov Chain Approximations to Semilinear Partial Differential Equations Driven by Poisson Measure Noise

We consider the stochastic model of water pollution, which mathematically can be written with a stochastic partial differential equation driven by Poisson measure noise. We use a stochastic particle Markov chain method to produce an implementable approximate solution. Our main result is the annealed law of large numbers establishing convergence in probability of our Markov chains to the solutio...

Difference Approximations to Partial Differential Equations