Counterexample to the convexity of level sets of solutions to the mean curvature equation
The convexity of level sets of solutions to the mean curvature equation is a long standing open problem. In this paper we give a counterexample to it.
Nodal solutions for a class of fourth-order two-point boundary value problems.
Hessian measures. II.
Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter.
A logarithmic Gauss curvature flow and the Minkowski problem
On weighted Hadamard-type singular integrals and their applications.
The mean curvature measure
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...
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