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.

We prove C 1,α regularity for a thin obstacle problem for the p-laplace equation. Due to the nonlinearity of the p-laplace operator we can not use the same methods used for the Laplace case, instead we use techniques developed by E. de Giorgi.

Abstract. The Volterra calculus is a simple and powerful pseudodifferential tool for inverting parabolic equations and it has also found many applications in geometric analysis. On the other hand, an important property in the theory of pseudodifferential operators is the asymptotic completeness, which allows us to construct parametrices modulo smoothing operators. In this paper we present new and fairly elementary proofs the asymptotic completeness of the Volterra calculus.

A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mappingz = √ z in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.