Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides

In this paper, we establish the existence of viscosity solutions of Hessian equations with singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The asymptotic results generalize those for Poisson equations and Monge-Ampère equations, and are more precise than obtained from Hopf lemma.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

The Calculus of Moving Surfaces and Laplace Eigenvalues on an Ellipse with Low Eccentricity

Our goal is to demonstrate the utility of the calculus of moving surfaces (CMS) in boundary variation problems. We discuss the relative advantages of the CMS compared to the alternative approach of interior variations. We illustrate the technique by calculating the two leading terms of a power series for the Laplace eigenvalues on an ellipse with semi-axes 1 + a and 1 + b, where a and b are small. We compare the CMS estimates with those obtained by the conventional finite element method with Richardson extrapolation. The comparison confirms the cubic rate of convergence for the CMS estimates.     

Thermal evolution of the first stage Cs(THF)1.75C24 graphitide,an example of segregation and decorrelation at low temperature

The thermal variation of the first stage ternary compound Cs(THF)_1.75C₂₄ has been investigated by powder neutron diffraction. Two phase transitions have been observed. The first one occurs at 248 K and can be interpreted as a three-dimensional segregation within the graphitic host. The second one starts at about 180 K and is continuous down to 50 K. This corresponds to a decorrelation of the graphen planes at low temperature. The origin of this transition could be the elastic constraint due to the kinks of the planes.     

Thermodynamic aspects of carbon redistribution during ageing and tempering of Fe–Ni–C alloys

Carbon redistribution is known to occur during martensite ageing. The two associated processes most discussed in the literature are spinodal decomposition and carbon segregation to defects. In order to elucidate the topic, the ageing and tempering of two Fe–Ni–C alloys have been characterised by means of atom probe tomography and synchrotron radiation diffraction. Upon ageing at room temperature, carbon redistribution is clearly observed, where the process of carbon segregation to defects appears to be most likely to occur. Nevertheless, the possibility of spinodal decomposition is not entirely discarded, and the current work presents a series of discussion points that challenge our current understanding of the thermodynamic of ferrite in steels.     

A variational approach for a generalized emden-fowler equation

Existence of positive solutions of singular boundary value problems related to Emden-Fowler equation is proved. A general minimization theorem in Sobolev spaces is applied.     

Criss‐cross amphiphilic polymers and analogous model systems

A new class of amphiphilic polymers carrying two pendant docosyl (C22) chains, located at periodic intervals that are separated by PEG chains of varying lengths, was synthesized via a simple melt‐transesterification polymerization, using dimethyl, 2,5‐didocosyloxyterephthalate as one of the monomers. DSC, variable temperature FT‐IR, and WAXS studies demonstrated that immiscibility between the pendant docosyl units and the backbone PEG segments drives their self‐segregation; this results in the crystallization of the pendant docosyl segments and the generation of a lamellar morphology with the alkyl segments and the PEG chains occupying alternate layers. Based on the study of model criss‐cross amphiphiles that resemble the polymer repeat unit, it is postulated that the chains reconfigure such that both the docosyl chains fold to one side of the terephthalate unit while the PEG segments form a loop on the other side; these chains then organize in a bilayer to form the lamellar structure. The simplicity of the synthesis and the rather unique properties of these polymers suggests that such a design could be translated to develop other interesting functional materials that could exploit the immiscibility‐driven microphase separation for the generation of sub‐10 nm domains; these could have potential applications, such as in membranes, solid polymer electrolyte formulations, and so forth. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 1554–1563     

Existence theorems for boundary value problems in difference equations

In this research, we study linear difference equations with constant coefficients subject to boundary conditions. Necessary and/or sufficient conditions for the existence of a unique solution will be established. The proofs of the existence and uniqueness theorems are established by means of special types of determinants called Mosaic Vandermonde determinants.     

Método de elementos de borde jerárquico basado en el árbol de Barnes-Hut aplicado a flujo reptante exterior

In this work, a hierarchical variant of a boundary element method and its use in Stokes flow around three-dimensional rigid bodies in steady regime is presented. The proposal is based on the descending hierarchical low-order and self-adaptive algorithm of Barnes-Hut, and it is used in conjunction with an indirect boundary integral formulation of second class, whose source term is a function of the undisturbed velocity. The solution field is the double layer surface density, which is modified in order to complete the eigenvalue spectrum of the integral operator. In this way, the rigid modes are eliminated and both a non-zero force and a non-null torque on the body could be calculated. The elements are low order flat triangles, and an iterative solution by generalized minimal residual (GMRES) is used. Numerical examples include cases with analytical solutions, bodies with edges and vertices, or with intricate shapes. The main advantage of the presented technique is the possibility of considering a greater number of degrees of freedom regarding traditional collocation methods, due to the decreased demand of main memory and the reduction in the computation times.