234 research outputs found
A priori estimates and application to the symmetry of solutions for critical -Laplace equations
We establish pointwise a priori estimates for solutions in
of equations of type , where
, \Delta_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla
u\big) is the -Laplace operator, and is a Caratheodory function with
critical Sobolev growth. In the case of positive solutions, our estimates allow
us to extend previous radial symmetry results. In particular, by combining our
results and a result of Damascelli-Ramaswamy, we are able to extend a recent
result of Damascelli-Merch\'an-Montoro-Sciunzi on the symmetry of positive
solutions in of the equation ,
where
A posteriori error bounds for discontinuous Galerkin methods for quasilinear parabolic problems
We derive a posteriori error bounds for a quasilinear parabolic problem,
which is approximated by the -version interior penalty discontinuous
Galerkin method (IPDG). The error is measured in the energy norm. The theory is
developed for the semidiscrete case for simplicity, allowing to focus on the
challenges of a posteriori error control of IPDG space-discretizations of
strictly monotone quasilinear parabolic problems. The a posteriori bounds are
derived using the elliptic reconstruction framework, utilizing available a
posteriori error bounds for the corresponding steady-state elliptic problem.Comment: 8 pages, conference ENUMATH 200
Boundedness of stable solutions to semilinear elliptic equations: a survey
This article is a survey on boundedness results for stable solutions to
semilinear elliptic problems. For these solutions, we present the currently
known estimates that hold for all nonlinearities. Such estimates
are known to hold up to dimension 4. They are expected to be true also in
dimensions 5 to 9, but this is still an open problem which has only been
established in the radial case
Boundedness of stable solutions to semilinear elliptic equations: a survey
This article is a survey on boundedness results for stable solutions to
semilinear elliptic problems. For these solutions, we present the currently
known estimates that hold for all nonlinearities. Such estimates
are known to hold up to dimension 4. They are expected to be true also in
dimensions 5 to 9, but this is still an open problem which has only been
established in the radial case
Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations
In this paper, we study fully non-linear elliptic equations in non-divergence
form which can be degenerate when "the gradient is small". Typical examples are
either equations involving the -Laplace operator or Bellman-Isaacs equations
from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci
estimate and we prove a Harnack inequality for viscosity solutions of such
degenerate elliptic equations.Comment: 27 pages. To appear in JD
Continuous dependence results for Non-linear Neumann type boundary value problems
We obtain estimates on the continuous dependence on the coefficient for
second order non-linear degenerate Neumann type boundary value problems. Our
results extend previous work of Cockburn et.al., Jakobsen-Karlsen, and
Gripenberg to problems with more general boundary conditions and domains. A new
feature here is that we account for the dependence on the boundary conditions.
As one application of our continuous dependence results, we derive for the
first time the rate of convergence for the vanishing viscosity method for such
problems. We also derive new explicit continuous dependence on the coefficients
results for problems involving Bellman-Isaacs equations and certain quasilinear
equation
- …