Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

$$\mathcal {H}^+$$-multivalued contractions and their application to homotopy theory

Kikkawa and Suzuki (Nonlinear Anal 69:2942–2949, 2008) and Kikkawa and Suzuki (Fixed Point Theory Appl, 2008, Art. ID 649749) proved some fixed point results that are generalizations of Kannan’s, Nadler’s and Suzuki’s fixed point theorems. Here, we present fixed point results of this kind for multivalued mappings in the setting of \(\mathcal {H}^+\)-metric spaces. The theorems provided allow upgrading of some known results which is shown by examples. Moreover, we give a homotopy result as an application of our main theorem.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

The elastic sphere under nonsymmetric loading

Existing solutions to boundary value problems arising from an elastic sphere subjected to a body force have been primarily restricted to axisymmetric, conservative loading. In this paper, a method for solving the displacement equations governing the static equilibrium of an elastic sphere subjected to an arbitrary body force and surface displacement is presented. The solutions are obtained in terms of three vector spherical harmonics and expressions for the displacement and stress fields are presented. Additionally, a short discussion indicating extension of these solutions to dynamic problems is included.This research was supported in part by an Organized Research Grant, Southwest Texas State University, 1979.     

Nonlinear Stability for Multidimensional Fourth-Order Shock Fronts

We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with only fourth-order regularization. Such equations arise, for example, in the study of thin films, for which planar waves correspond to fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive, and undercompressive, planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, Münch, and Shearer indicates that undercompressive waves can be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth-order convection–regularization equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general spectral conditions, such as appear to hold for shock fronts arising in our motivating thin films equations, compressive waves are stable for all dimensions d≧2 and undercompressive waves are stable for dimensions d≧3. (In the special case d=1, compressive waves are stable under a very general spectral condition.) We also consider an alternative spectral criterion (valid, for example, in the case of constant-coefficient regularization), for which we can establish nonlinear stability for compressive waves in dimensions d≧3 and undercompressive waves in dimensions d≧5. The case of stability for undercompressive waves in the thin films equations for the critical dimensions d=1 and d=2 remains an interesting open problem.     

Integral equation theory for the study of structure and thermodynamics and scaling-aggregation of droplets within Pickering emulsions versus the salt-concentration

In this work, we aim at the study of the statistical and thermal properties of the oil-in-water Pickering emulsions which are stabilized by strongly adsorbed charged nanoparticles on the surfaces of their droplets. In this study, we have adopted a pair-potential of Sogami-Ise type which is the sum of a repulsive part and an attractive one; and depends, in particular, on the salt-concentration. In this work, we were interested in two kinds of problems, namely (1) the determination of the structural and thermal properties of Pickering emulsions, and (2) the investigation of the scaling of cluster aggregation of the oil-droplets, as the salt-concentration is varied, keeping the other pertinent factors (surface-charge of oil-droplets, their size and density and bath temperature) fixed. For the two problems, we regarded the oil-droplets as monodisperse small spheres assimilated to charged soft-colloids. First, we have computed all of the structural properties (through the radial-distribution-function) versus the salt-concentration using the Integral-Equation Theory with the Hybridized-Mean-Spherical Approximation. Second, using the same numerical appoach, we have determined the thermal properties (through pressure, internal energy and isothermal compressibility). Finally, we have demonstraded that the oil-droplets aggregate has a fractal structure, with the help of an elaborated scaling model which was found to be in good agreement with Integral-Equation Theory.     

A normalized solitary wave solution of the Maxwell-Dirac equations

We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.     

Probabilistic modeling and global sensitivity analysis for CO2 storage in geological formations: a spectral approach

This work focuses on the simulation of CO₂ storage in deep underground formations under uncertainty and seeks to understand the impact of uncertainties in reservoir properties on CO₂ leakage. To simulate the process, a non-isothermal two-phase two-component flow system with equilibrium phase exchange is used. Since model evaluations are computationally intensive, instead of traditional Monte Carlo methods, we rely on polynomial chaos (PC) expansions for representation of the stochastic model response. A non-intrusive approach is used to determine the PC coefficients. We establish the accuracy of the PC representations within a reasonable error threshold through systematic convergence studies. In addition to characterizing the distributions of model observables, we compute probabilities of excess CO₂ leakage. Moreover, we consider the injection rate as a design parameter and compute an optimum injection rate that ensures that the risk of excess pressure buildup at the leaky well remains below acceptable levels. We also provide a comprehensive analysis of sensitivities of CO₂ leakage, where we compute the contributions of the random parameters, and their interactions, to the variance by computing first, second, and total order Sobol’ indices.     

Oscillatory properties of first order linear functional differential equations

Oscillatory properties of retarded and advanced functional differential equations are investigated.In the first part, the study concerns equations with piecewise constant arguments, which found applications in certain biomedical problems. Then, results of some authors are generalized for general equations with many argument deviations. Finally, applications are given to equations with linear transformations of the argument.