Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userModeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity
98
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.
rate research
Read More
We study variational problems involving nonlocal supremal functionals $L^infty(Omega;mathbb{R}^m) i umapsto {rm ess sup}_{(x,y)in Omegatimes Omega} W(u(x), u(y)),$ where $Omegasubset mathbb{R}^n$ is a bounded, open set and $W:mathbb{R}^mtimesmathbb{R}^mto mathbb{R}$ is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for $L^infty$-weak$^ast$ lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. Whether the same statement holds in the related context of double-integral functionals is currently still open. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak$^ast$ closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.
A multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for multicomponent mixtures, where compressible, hydrodynamic, and thermodynamic nonequilibrium effects are taken into account. It allows the specific heat ratio and the Prandtl number to be adjustable, and is suitable for both low and high speed fluid flows. From the physical side, besides being consistent with the multicomponent Navier-Stokes equations, Ficks law and Stefan-Maxwell diffusion equation in the hydrodynamic limit, the DBM provides more kinetic information about the nonequilibrium effects. The physical capability of DBM to describe the nonequilibrium flows, beyond the Navier-Stokes representation, enables the study of the entropy production mechanism in complex flows, especially in multicomponent mixtures. Moreover, the current kinetic model is employed to investigate nonequilibrium behaviors of the compressible Kelvin-Helmholtz instability (KHI). It is found that, in the dynamic KHI process, the mixing degree and fluid flow are similar for cases with various thermal conductivity and initial temperature configurations. Physically, both heat conduction and temperature exert slight influences on the formation and evolution of the KHI.
In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a $Gamma$-convergence result on the interaction energy as the number of particles tends to infinity. As the leading application, we consider $n$ edge dislocations in multiple slip systems. Since the interaction potential of dislocations has a logarithmic singularity at zero with a sign that depends on the orientation of the slip systems, the interaction energy is unbounded from below. To make the minimization problem of this energy meaningful, we follow the common approach to regularise the interaction potential over a length-scale $delta > 0$. The novelty of our result is that we leave the emph{type} of regularisation general, and that we consider the joint limit $n to infty$ and $delta to 0$. Our result shows that the limit behaviour of the interaction energy is not affected by the type of the regularisation used, but that it may depend on how fast the emph{size} (i.e., $delta$) decays as $n to infty$.
Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with Bounded Deformation. Mild additional continuity assumptions are required on the functionals.
The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ left{ begin{array}{l} - Delta_1 u +xi frac{u}{|u|} =lambda |u|^{q-2}u+|u|^{1^*-2}u, quadtext{in }Omega, u=0, quadtext{on } partialOmega. end{array} right. $$ where $Omega$ is a smooth bounded domain in $mathbb{R}^N$, $N geq 2$ and $xi in{0,1}$. Moreover, $lambda > 0$, $q in (1,1^*)$ and $1^*=frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $xi=1$, $Omega = {x in mathbb{R}^N,:,r < |x| < r+1}$, $Ngeq 2$, $N ot = 3$ and $r > 0$. In the second one, $Omega$ is a smooth bounded domain, $xi=0$, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a $C^1$ functional with a convex lower semicontinuous functional.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
