Archive for Rational Mechanics and Analysis最新文献

In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem

where (mathcal {M}) ranges over all n-dimensional manifolds in (mathbb {R}^{n+1}) with a prescribed boundary, (nabla _{mathcal {M}}H) is the tangential gradient along (mathcal {M}) of the mean curvature H of (mathcal {M}) and dA is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results in the case of graphs. These are the first analytic results available for this problem.

We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and 2d-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples—in particular, we improve in an essentially optimal fashion Marcellini’s original results (Marcellini in Arch Rat Mech Anal 105:267–284, 1989).

In this paper we consider the steepest descent (L^2)-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class (C^2) or embedded of class (W^{2,2}) bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.

We prove that 2-dimensional Q-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions.

The subject of so-called objective derivatives in Continuum Mechanics has a long history and has generated varying views concerning their true mathematical interpretation. Several attempts have been made to provide a mathematical definition that would at least partially unify the existing notions. In this paper, we demonstrate that, under natural assumptions, all objective derivatives correspond to covariant derivatives on the infinite-dimensional manifold (textrm{Met}(mathcal {B})) of Riemannian metrics on the body. Furthermore, a natural Leibniz rule enables canonical extensions from covariant to contravariant tensor fields and vice versa. This makes the sometimes-used distinction between objective derivatives of “Lie type” and “co-rotational type” unnecessary. For an exhaustive list of objective derivatives found in the literature, we exhibit the corresponding covariant derivative on (textrm{Met}(mathcal {B})).

A 7-dimensional area-minimizing embedded hypersurface (M^7) will in general have a discrete singular set, and the same is true if M is locally stable provided ({mathcal {H}}^6(textrm{sing}M) = 0). We show that if (M_i^7) is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then (M_i rightarrow M) can limit to a singular (M^7) with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence (i') with controlled bi-Lipschitz maps (phi _{i'}) taking (phi _{i'}(M_{1'}) = M_{i'}). As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces M in a closed Riemannian 8-manifold ((N^8, g)) with a priori bounds ({mathcal {H}}^7(M) leqq Lambda ) and (textrm{index}(M) leqq I) divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric g to vary, or M to be singular.

We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.

Whereas many different models exist in mathematics and physics for the ground states of non-relativistic crystals, the relativistic case has been much less studied, and we are not aware of any mathematical result on a fully relativistic treatment of crystals. In this paper, we introduce a mean-field relativistic energy for crystals in terms of periodic density matrices. This model is inspired both from a recent definition of the Dirac–Fock ground state for atoms and molecules, due to one of us, and from the non-relativistic Hartree–Fock model for crystals. We prove the existence of a ground state when the number of electrons per cell is not too large.

We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a stable way, from the boundary measurements, by applying a linearization scheme and Carleman estimates for the linear transport equations. We establish results in both Euclidean and general geometry settings.