We consider p independent Brownian motions in \input amssym ${\Bbb R}^d$. We assume that p ≥ 2 and p (d − 2) < d. Let ℓ_t denote the intersection measure of the p paths by time t, i.e., the random measure on \input amssym ${\Bbb R}^d$ that assigns to any measurable set \input amssym $A \subset {\Bbb R}^d$ the amount of intersection local time of the motions spent in A by time t. Earlier results of X. Chen derived the logarithmic asymptotics of the upper tails of the total mass \input amssym $\ell _t \left({{\Bbb R}^d } \right)$ as t → ∞. In this paper, we derive a large-deviation principle for the normalized intersection measure t^−pℓ_t on the set of positive measures on some open bounded set \input amssym $B \subset {\Bbb R}^d$ as t → ∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalized occupation times measures of the p motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure.

A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set . This extends earlier studies on the intersection measure by König and Mörters. © 2012 Wiley Periodicals, Inc.

In this paper, we study the compactness in L of the semigroup (St)_t≥0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of St for each t > 0 was established by P. D. Lax. Upper estimates for the Kolmogorov e-entropy of the image of bounded sets in L¹ n L^∞ through S_t were given by C. De Lellis and F. Golse. Here we provide lower estimates on this e-entropy of the same order as the one established by De Lellis and Golse, thus showing that such an e-entropy is of size ≈ 1/ε. Moreover, we extend these estimates of compactness to the case of convex balance laws. © 2012 Wiley Periodicals, Inc.

The elastoplastic quasi-static evolution of a multiphase material—a material with a pointwise varying yield surface and elasticity tensor, together with interfaces between the phases—is revisited in the context of conservative globally minimizing movements. Existence is shown, and classical evolutions are recovered under natural constraints on the plastic dissipation potential. Special attention is paid to the interfaces where the correct dissipation has to be enforced on the interfaces. Further, the evolution is shown to be a limit of that obtained for a model with linear isotropic hardening as the hardening becomes vanishingly small. The duality between plastic strains and admissible stresses is also revisited for Lipschitz boundaries, and its role in deriving a classical evolution is circumscribed. © 2012 Wiley Periodicals, Inc.

We consider the minimization of in a perforated domain of among maps that are incompressible (det ) and invertible, and satisfy a Dirichlet boundary condition u = g on ∂Ω. If the volume enclosed by g(∂Ω) is greater than |Ω|, any such deformation u is forced to map the small holes B_ε(a_i) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of and the model is suited, therefore, for an asymptotic analysis (v₁,…, v_M denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg-Landau theory, we obtain estimates for the “renormalized” energy

showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a₁,…,a_M, and on the distance from these points to the outer boundary ∂Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.

We consider the Cauchy problem for quadratic nonlinear Klein-Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.-M. Delort, D. Fang, and R. Xue (J. Funct. Anal. 211 (2004), no. 2, 288–323), we show the global existence of asymptotically free solutions if the initial data are sufficiently small in some weighted Sobolev space. Our proof is based on an algebraic characterization of nonlinearities satisfying the null condition. © 2012 Wiley Periodicals, Inc.

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar θ by , where and is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch-type solutions. The second family is a dissipative active scalar equation with , which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc.

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDMis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold embedded in , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold. © 2012 Wiley Periodicals, Inc.

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q-regular measure, where Q > 1, that supports a local L²-Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L^∞-norm of the gradient Du can be bounded by the Lorentz norm L^Q,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on . © 2011 Wiley Periodicals, Inc.

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is C^2,α on the complement of a closed set of Hausdorff dimension at most ϵ less than the dimension. The equation is assumed to be C¹, and the constant ϵ > 0 depends only on the dimension and the ellipticity constants. The argument combines the W^2,ϵ estimates of Lin with a result of Savin on the C^2,α regularity of viscosity solutions that are close to quadratic polynomials. © 2012 Wiley Periodicals, Inc.