2,243 research outputs found
Fully Degenerate Monge Amp\'ere Equations
In this paper, we consider the following nonlinear eigenvalue problem for the
Monge-Amp\'ere equation: find a non-negative weakly convex classical solution
satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in }
f=\vp \quad &text{on } {cases} {equation*} for a strictly
convex smooth domain and . When contains a
convex domain, we find a classical solution which is smooth on
and whose free boundary is also smooth
System of Porous Medium Equations
We investigate the evolution of population density vector,
, of -species whose diffusion is
controlled by its absolute value . More precisely we
study the properties and asymptotic large time behaviour of solution
of degenerate parabolic system
\begin{equation*}
\left(u^i\right)_t=\nabla\cdot\left(\left|\bold{u}\right|^{m-1}\nabla
u^i\right) \qquad \mbox{for and }. \end{equation*} Under
some regularity assumption, we prove that the function which describes
the population density of -th species with population converges to
in space
with two different approaches where is
the Barenblatt solution of the porous medium equation with -mass
. \indent As an application of
the asymptotic behaviour, we establish a suitable harnack type inequality which
makes the spatial average of under control by the value of at one
point. We also find an 1-directional travelling wave type solutions and the
properties of solutions which has travelling wave behaviour at infinity.Comment: 30 page
H\"older regularity and Uniqueness theorem on weak solutions to the degenerate Keller-Segel system
In this paper, we present local H\"older estimates for the degenerate
Keller-Segel system
\eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} below in the range
of and before a blow-up of solutions. To deal with difficulties
caused by the degeneracy of the operator, we find uniform estimates depending
sup-norm of the density function and modified the energy estimates and
intrinsic scales considered in Porous Medium Equation. As its application, the
uniqueness of weak solution to
\eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} is also showed in
the class of H\"older continuous functions by proving -contraction in this
class.Comment: 23 page
Geometric Properties of Gelfand's Problems with Parabolic Approach
We consider the asymptotic profiles of the nonlinear parabolic flows
(e^{u})_{t}= \La u+\lambda e^u to show the geometric properties of the
following elliptic nonlinear eigenvalue problems known as a Gelfand's problem:
\begin{equation*} \begin{split} \La \vp &+ \lambda e^{\vp}=0, \quad
\vp>0\quad\text{in }\\ \vp&=0\quad\text{on } \end{split}
\end{equation*} posed in a strictly convex domain \Omega\subset\re^n. In
this work, we show that there is a strictly increasing function such
that f^{-1}(\vp(x)) is convex for , i.e., we
prove that level set of \vp is convex. Moreover, we also present the boundary
condition of \vp which guarantee the -convexity of solution \vp.Comment: 16 page
Higher Order Convergence Rates in Theory of Homogenization III: viscous Hamilton-Jacobi Equations
In this paper, we establish the higher order convergence rates in periodic
homogenization of viscous Hamilton-Jacobi equations, which is convex and grows
quadratically in the gradient variable. We observe that although the nonlinear
structure governs the first order approximation, the nonlinear effect is
absorbed as an external source term of a linear equation in the second and
higher order approximation. Moreover, we find that the geometric shape of the
initial data has to be chosen carefully according to the effective Hamiltonian,
in order to achieve the higher order convergence rates
The Green function for the Stokes system with measurable coefficients
We study the Green function for the stationary Stokes system with bounded
measurable coefficients in a bounded Lipschitz domain , . We construct the Green function in under the
condition that weak solutions of the system enjoy interior H\"older
continuity. We also prove that holds, for example, when the
coefficients are . Moreover, we obtain the global pointwise
estimate for the Green function under the additional assumption
that weak solutions of Dirichlet problems are locally bounded up to the
boundary of the domain. By proving a priori -estimates for Stokes systems
with coefficients on a Reifenberg domain, we verify that
is satisfied when the coefficients are and
is a bounded domain.Comment: 35 pages; part of Section 2.2 is revised; accepted in Communications
on Pure and Applied Analysi
Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems
We study the local H\"older continuity and the asymptotic behaviour of
solution, , of the degenerate system
\begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad
\text{for and } \end{equation*} which describes the
populations density of -species whose diffusion is determined by their total
population density . For the local H\"older continuity, we
adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and
Dibenedetto. Under some regularity conditions, we also prove that the
population density function of -th species with the population
converges in to as where is the Barenblatt profile of the standard porous
medium equation with mass . As a consequence of
asymptotic behaviour, it is shown that each density function becomes a concave
function after a finite time.Comment: 37
Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels
In this paper, we consider fully nonlinear integro-differential equations
with possibly nonsymmetric kernels. We are able to find different versions of
Alexandroff-Backelman-Pucci estimate corresponding to the full class
\cS^{\fL_0} of uniformly elliptic nonlinear equations with
(subcritical case) and to their subclass \cS^{\fL_0}_{\eta} with
. We show that \cS^{\fL_0}_{\eta} still includes a large
number of nonlinear operators as well as linear operators. And we show a
Harnack inequality, H\"older regularity, and -regularity of the
solutions by obtaining decay estimates of their level sets in each cases
Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders
We establish the generalized Evans--Krylov and Schauder type estimates for
nonlocal fully nonlinear elliptic equations with rough kernels of variable
orders. In contrast to the fractional Laplacian type operators having a fixed
order of differentiability , the operators under
consideration have variable orders of differentiability. Since the order is not
characterized by a single number, we consider a function describing
the variable orders of differentiability, which is allowed to oscillate between
two functions and for some . By introducing the generalized H\"older spaces, we provide
estimates that generalizes the standard Evans--Krylov and
Schauder type estimates
The Evans-Krylov theorem for nonlocal parabolic fully nonlinear equations
In this paper, we prove the Evans-Krylov theorem for nonlocal parabolic fully
nonlinear equations
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