We study an elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Holder continuity of weak solutions and obtain pointwise bounds for Greens functions subject to Dirichlet or Neumann condition.
Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Holder semi-norms not with respect to all, but only with respect to some of the independent
variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.
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, r
egularity of the generalized solution of such a problem remained unclear. We prove that the generalized solution of the problem is indeed a classical solution.
We establish existence and pointwise estimates of fundamental solutions and Greens matrices for divergence form, second order strongly elliptic systems in a domain $Omega subseteq mathbb{R}^n$, $n geq 3$, under the assumption that solutions of the sy
stem 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.
We study Greens 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 Greens matrices.
We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
We establish existence and various estimates of fundamental matrices and Greens matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an inter
ior H{o}lder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.
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.