12,930 research outputs found
The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
The vanishing moment method was introduced by the authors in [37] as a
reliable methodology for computing viscosity solutions of fully nonlinear
second order partial differential equations (PDEs), in particular, using
Galerkin-type numerical methods such as finite element methods, spectral
methods, and discontinuous Galerkin methods, a task which has not been
practicable in the past. The crux of the vanishing moment method is the simple
idea of approximating a fully nonlinear second order PDE by a family
(parametrized by a small parameter \vepsi) of quasilinear higher order (in
particular, fourth order) PDEs. The primary objectives of this book are to
present a detailed convergent analysis for the method in the radial symmetric
case and to carry out a comprehensive finite element numerical analysis for the
vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract
methodological and convergence analysis frameworks of conforming finite element
methods and mixed finite element methods are first developed for fully
nonlinear second order PDEs in general settings. The abstract frameworks are
then applied to three prototypical nonlinear equations, namely, the
Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the
infinity-Laplacian equation. Numerical experiments are also presented for each
problem to validate the theoretical error estimate results and to gauge the
efficiency of the proposed numerical methods and the vanishing moment
methodology.Comment: 141 pages, 16 figure
Functional Inequalities for Particle Systems on Polish Spaces
Various Poincare-Sobolev type inequalities are studied for a
reaction-diffusion model of particle systems on Polish spaces. The systems we
consider consist of finite particles which are killed or produced at certain
rates, while particles in the system move on the Polish space interacting with
one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we
call reaction-diffusion Dirichlet form, consists of two parts: the diffusion
part induced by certain Markov processes on the product spaces
which determine the motion of particles, and the reaction part induced by a
-process on and a sequence of reference probability measures,
where the -process determines the variation of the number of particles and
the reference measures describe the locations of newly produced particles. We
prove that the validity of Poincare and weak Poincare inequalities are
essentially due to the pure reaction part, i.e. either of these inequalities
holds if and only if it holds for the pure reaction Dirichlet form, or
equivalently, for the corresponding -process. But under a mild condition,
stronger inequalities rely on both parts: the reaction-diffusion Dirichlet form
satisfies a super Poincare inequality (e.g. the log-Sobolev inequality) if and
only if so do both the corresponding -process and the diffusion part.
Explicit estimates of constants in the inequalities are derived. Finally, some
specific examples are presented to illustrate the main results.Comment: 22 pages, BiBoS-Preprint no. 04-08-153, to appear in Potential
Analysi
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