Regularity of the free boundary in the biharmonic obstacle problem

In this article we use a flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible domain, we derive the $$C^{1,\alpha }$$C1,α-regularity of the free boundary in a small ball centred at the origin. From the $$C^{1,\alpha }$$C1,α-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $$ C^{3,\alpha }$$C3,α up to the free boundary, and therefore $$C^{2,1}$$C2,1. In the end we study an example, showing that in general $$ C^{2,\frac{1}{2}}$$C2,12 is the best regularity that a solution may achieve in dimension $$n \ge 2$$n≥2.

Reidemeister classes for coincidences between sections of a fiber bundle

Let $s_0,f_0$ be two sections of a fiber bundle $q: E\to B$ and the coincidence set $\Gamma(s_0,f_0)\neq \emptyset$. We consider the following question: Is there $s_0\simeq_B s_1$ (by the homotopies which cover the constant homotopy $\overline{I}_B$ on the basic space) such that $\Gamma(s_1, f_0)=\emptyset$? If $b_0\in \Gamma(s_0,f_0)$ and $F_0=q^{-1}(b_0)$ is the typical fiber, in this context we can use the homotopy lifting extension propriety of the fibration $q$ to obtain homotopies over $B$. When we make this and the basic point are fixed we can use the elements $s_0(\beta), f_0(\beta^{-1})$ where $\beta \in\pi_1(B,b_0)$ and the elements $\gamma\in \pi_1(F_0,e_0)$. So we will introduce the algebraic classes of Reidemeister relative to the subgroup $\pi_1(F_0,e_0)$. When the basic points are not fixed we need to consider the classes $[\til{s}]_L$ of lifting of $s_0$ defined on the universal covering $\til{B}$ to $\til{E}$. The present work relates the lifting classes $[\til{s}]_L$ of $s_0$ and the algebraic relative Reidemeister classes $R_A(s_0,f_0; \pi_1(F_0,e_0).$

Some remarks on the coincidence set for the Signorini problem

We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data.

Approximation of a simply supported plate with obstacle

We discuss an algorithm for the solution of variational inequalities associated with simply supported plates in contact with a rigid obstacle. Our approach has a fixed domain character, uses just linear equations and approximates both the solution and the corresponding coincidence set. Numerical examples are also provided.

A direct algorithm in some free boundary problems

In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one- and two- phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The approximating coincidence set is explicitly computed and it converges in the Hausdorff-Pompeiu sense to the searched geometry. In the numerical examples, the algorithm has a very fast convergence and the obtained solutions (including the free boundaries) are accurate.

Chemical Transformation of Crystalline Hafnium Tetrafluoride Studied by Perturbed Angular Correlation Spectroscopy

The chemical transformation of the trihydrate hafnium tetrafluoride crystal has been studied with varying temperature using the time-differential perturbed angular correlation technique. The 133 - 482 keV γ -γ cascade of 181Ta after the β −-decay of 181Hf has been selected and a four detector BaF2-BaF2 coincidence set up has been used for measurements. The crystal was produced by evaporating a solution of HfF62−complex in HF at room temperature. Contrary to the earlier report, it has been found that the trihydrate hafnium tetrafluoride compound dehydrates directly to HfF4 without producing any intermediate monohydrate and present results do not support the earlier idea that two water molecules of HfF4·3H2O are loosely bound. Present investigations exhibit a superheated state for the hafnium tetrafluoride crystal. In dehydrated HfF4, two different Hf sites have been observed which suggests two different structures for the anhydrous HfF4.