Abstract Given a smooth closed embedded self-shrinker S with index I in $$\mathbb {R}^{n}$$ R n , we construct an I-dimensional family of complete translators polynomially asymptotic to $$S\times \mathbb {R}$$ S × R at infinity, which answers a long-standing question by Ilmanen. We further prove that $$\mathbb {R}^{n+1}$$ R n + 1 can be decomposed in many ways into a one-parameter family of closed sets $$\coprod _{a\in \mathbb {R}} T_a$$ ∐ a ∈ R T a , and each closed set $$T_a$$ T a contains a complete translator asymptotic to $$S\times \mathbb {R}$$ S × R at infinity. If the closed set $$T_a$$ T a fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens.

Abstract We deal with the inverse problem of reconstructing acoustic material properties or/and external sources for the time-domain acoustic wave model. The traditional measurements consist of repeated active (or passive) interrogations, such as the Dirichlet-Neumann map, or point sources with source points varying outside of the domain of interest. It is reported in the existing literature that based on such measurements, one can recover some (but not all) of the three parameters: mass density, bulk modulus or the external source term. In this work, we first inject isolated small-scales bubbles into the region of interest and then measure the generated pressure field at a single point outside, or at the boundary, of this region. Then we repeat such measurements by moving the bubble to scan the region of interest. Using such measurements, we show that If either the mass density or the bulk modulus is known then we can simultaneously reconstruct the other one and the source term. If the source term is known at the initial time, precisely we assume to know its first non vanishing time-derivative, at the initial time, then we reconstruct simultaneously the two parameters, namely the mass density with the bulk modulus and eventually the source function. Here, the source term, which is space-time dependent, can be active (and hence known) or passive (and unknown). It is worth mentioning that in the induced inverse problem, we use measurements with $$4=3+1$$ 4 = 3 + 1 dimensions (3 in space and 1 in time) to recover 2 coefficients of 3 spatial dimensions, i.e. the mass density and the bulk modulus and the 4 = 3 + 1 dimensional source function. In addition, the result is local, meaning that we do reconstruction in any subpart, of the domain of interest, we want.

Abstract In the limit of vanishing lattice spacing we provide a rigorous variational coarse-graining result for a next-to-nearest neighbor lattice model of a simple crystal. We show that the $$\Gamma $$ Γ -limit of suitable scaled versions of the model leads to an energy describing a continuum mechanical model depending on partial dislocations and stacking faults. Our result highlights the necessary multiscale character of the energies setting the groundwork for more comprehensive models that can better explain and predict the mechanical behavior of materials with complex defect structures.