Searcharxiv⌕ Search

arXiv subjects

Seick Kim

Publications and source records attributed to Seick Kim.

47 records · Page 3Linked to original sources
arXiv2011

Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media

We obtain sharp $L^p$ and Hölder estimates for the Neumann function of the operator $\nabla \cdot γ\nabla - ik$ on a bounded domain. We also obtain quantitative description of its singularity. We then apply these estimates to quantitative photo-acoustic imaging in inhomogeneous media. The problem is to reconstruct the optical absorption coefficient of a diametrically small anomaly from the absorbed energy density.

math.AP↗
arXiv2010

Green's functions for parabolic systems of second order in time-varying domains

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and 1/2-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder continuity estimate. In particular, our results apply to complex perturbations of a single real equation.

math.AP↗
arXiv2010

Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains

We establish global Gaussian estimates for the Green's matrix of divergence form, second order parabolic systems in a cylindrical domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder estimate. From these estimates, we also derive global estimates for the Green's matrix for elliptic systems with bounded measurable coefficients in two dimensional domains. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.

math.AP↗
arXiv2010

Global pointwise estimates for Green's matrix of second order elliptic systems

We establish global pointwise bounds for the Green's matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green's matrix. In the scalar case, such an estimate is a consequence of De Giorgi-Moser-Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In the vectorial case, one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.

math.AP↗
arXiv2008

On a degenerate parabolic equation arising in pricing of Asian options

We study a certain one dimensional, degenerate parabolic partial differential equation with a boundary condition which arises in pricing of Asian options. Due to degeneracy of the partial differential operator and the non-smooth boundary condition, regularity of the generalized solution of such a problem remained unclear. We prove that the generalized solution of the problem is indeed a classical solution.

math.AP↗
arXiv2007

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 $Ω\subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.

math.AP↗
arXiv2007

On the Green's matrices of strongly parabolic systems of second order

We establish existence and various estimates of fundamental matrices and Green's matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior Hölder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.

math.AP↗
arXiv2007

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 $\mathbb{R}^n$. In particular, in the case when $n=2$ 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.

math.AP↗