Hayk Mikayelyan

The zero level set for a certain weak solution, with applications to the Bellman equations

We will prove a partial regularity result for the zero level set of weak solutions to div(B∇u) = 0, where B = B(u) = I + (A− I)χ{u<0}, where I is the identity matrix and the eigenvalues of A are strictly positive and bounded. We will apply this to describe the regularity of solutions to the Bellman equations.

An Elliptic Optimal Control Problem and its Two Relaxations

In this note, we consider a control theory problem involving a strictly convex energy functional, which is not Gâteaux differentiable. The functional came up in the study of a shape optimization problem, and here we focus on the minimization of this functional. We relax the problem in two different ways and show that the relaxed variants can be solved by applying some recent results on two-phase obstacle-like problems of free boundary type. We derive an important qualitative property of the solutions, i.e., we prove that the minimizers are three-valued, a result which significantly reduces the search space for the relevant numerical algorithms.

Stabilization Technique Applied To Curve Shortening Flow in the Plane

The method proposed by T. I. Zelenjak is applied to the mean curvature flow in the plane. A new type of monotonicity formula for star-shaped curves is obtained.