60 research outputs found
Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients
Auscher, McIntosh and Tchamitchian studied the heat kernels of second order
elliptic operators in divergence form with complex bounded measurable
coefficients on . In particular, in the case when they
obtained Gaussian upper bound estimates for the heat kernel without imposing
further assumption on the coefficients. We study the fundamental solutions of
the systems of second order parabolic equations in the divergence form with
bounded, measurable, time-independent coefficients, and extend their results to
the systems of parabolic equations
Regularity of a degenerate parabolic equation appearing in Vecer's unified pricing of Asian options
Vecer derived a degenerate parabolic equation with a boundary condition
characterizing the price of Asian options with generally sampled average. It is
well understood that there exists a unique probabilistic solution to such a
problem but it remained unclear whether the probabilistic solution is a
classical solution. We prove that the probabilistic solutions to Vecer's PDE
are regular.Comment: 6 page
The Green function estimates for strongly elliptic systems of second order
We establish existence and pointwise estimates of fundamental solutions and
Green's matrices for divergence form, second order strongly elliptic systems in
a domain , , under the assumption that
solutions of the system satisfy De Giorgi-Nash type local H\"{o}lder continuity
estimates. In particular, our results apply to perturbations of diagonal
systems, and thus especially to complex perturbations of a single real
equation.Comment: bibliography correcte
Neumann functions for second order elliptic systems with measurable coefficients
We study Neumann functions for divergence form, second order elliptic systems
with bounded measurable coefficients in a bounded Lipschitz domain or a
Lipschitz graph domain. We establish existence, uniqueness, and various
estimates for the Neumann functions under the assumption that weak solutions of
the system enjoy interior H\"older continuity. Also, we establish global
pointwise bounds for the Neumann functions under the assumption that weak
solutions of the system satisfy a certain natural local boundedness estimate.
Moreover, we prove that such a local boundedness estimate for weak solutions of
the system is in fact equivalent to the global pointwise bound for the Neumann
function. We present a unified approach valid for both the scalar and the
vectorial cases.Comment: 23 pages, 0 figure; accepted in Trans. Amer. Math. So
Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
We study Green's matrices for divergence form, second order strongly elliptic
systems with bounded measurable coefficients in two dimensional domains. We
establish existence, uniqueness, and pointwise estimates of the Green's
matrices.Comment: 18 page
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