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.

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ballWe prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data f and its first and second derivatives vanish at the touch-point.

Abstract. This paper concerns Hopf's boundary point lemma, in certain C 1,Dini -type domains, for a class of singular/degenerate PDE-s, including p-Laplacian. Using geometric properties of levels sets for harmonic functions in convex rings, we construct sub-solutions to our equations that play the role of a barrier from below. By comparison principle we then conclude Hopf's lemma.

In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants K (Korn's constant) in the inequalities depend on the domain thickness h according to a power rule K = Ch α , where C > 0 and α ∈ R are constants independent of h and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics h α is optimal as h → 0. The choice of the weights is motivated by several factors, in particular a spacial case occurs when making Cartesian to polar change of variables in two dimensions.

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.Date: 1/DEC/2014.