32,988 research outputs found
Nonlinear elliptic equations with high order singularities
We study non-variational degenerate elliptic equations with high order
singular structures. No boundary data are imposed and singularities occur along
an {\it a priori} unknown interior region. We prove that positive solutions
have a universal modulus of continuity that does not depend on their infimum
value. We further obtain sharp, quantitative regularity estimates for
non-negative limiting solutions.Comment: Revise
Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential
Equation which originated in geometric surface theory, and has been applied in
dynamic meteorology, elasticity, geometric optics, image processing and image
registration. Solutions can be singular, in which case standard numerical
approaches fail. In this article we build a finite difference solver for the
Monge-Ampere equation, which converges even for singular solutions. Regularity
results are used to select a priori between a stable, provably convergent
monotone discretization and an accurate finite difference discretization in
different regions of the computational domain. This allows singular solutions
to be computed using a stable method, and regular solutions to be computed more
accurately. The resulting nonlinear equations are then solved by Newton's
method. Computational results in two and three dimensions validate the claims
of accuracy and solution speed. A computational example is presented which
demonstrates the necessity of the use of the monotone scheme near
singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added
coment
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