Haigang Li

Gradient estimates for parabolic systems from composite material

In this paper, we derive W1,∞ and piecewise C1,α estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise Hölder continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56: 892–925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.

Gradient Estimates for Solutions of the Lamé System with Partially Infinite Coefficients

We establish upper bounds on the blow up rate of the gradients of solutions of the Lamé system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero.

On the exterior Dirichlet problem for Hessian equations

In this paper, we establish a theorem on the existence of the solutions of the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity. This extends a result of Caffarelli and Li in [3] for the MongeAmpere equation to Hessian equations.

Optimal Estimates for the Conductivity Problem by Green’s Function Method

We study a class of second-order elliptic equations of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. We establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions. These extend the known results in the literature and answer open problem (b) proposed by Li and Vogelius (2000) for the isotropic conductivity problem. We also obtain more interesting higher-order derivative estimates, which answers open problem (c) of Li and Vogelius (2000). It is worth pointing out that the equations under consideration in this paper are non-homogeneous.

Optimal Estimates for the Perfect Conductivity Problem with Inclusions Close to the Boundary

When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a “bigger” convex domain containing the inclusion, the electric field can be arbitrarily large. We establish both the pointwise upper bound and the lower bound of the gradient estimate for this perfect conductivity problem by using the energy method. These results give the optimal blowup rates of an electric field for conductors with arbitrary shape and in all dimensions. A particular case when a circular inclusion is close to the boundary of a circular matrix domain in dimension two is studied earlier by Ammari et al. [J. Math. Pures Appl. (9), 88 (2007), pp. 307--324]. From the view of methodology, the technique we develop in this paper is significantly different from the previous one restricted to the circular case, which allows us further investigate the general elliptic equations with divergence form.