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 consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional c
urvature of the underlying manifold is nonnegative. Later, Kim improved Cabres result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.
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.
