329 research outputs found
k-quasiconvexity reduces to quasiconvexity
The relation between quasi-convexity and k-quasiconvexity (k greater than or equal to 2) is investigated. It is shown that every smooth strictly k-quasi-convex integrand with p-growth at infinity, p > 1, is the restriction to kth-order symmetric tensors of a quasiconvex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semicontinuity results for kth-order variational problems are deduced as corollaries of well-known first-order theorems. This generalizes a previous work by Dal Maso et al., in which the case where k = 2 was treated
Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach
A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation
of constant unloading response, the result contained in [G. Dal Maso, C. Zanini: Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. Roy. Soc. Edinburgh Sect. A, 137A (2007), 253–279.] is recovered. In this case, the convergence of the discrete time approximations is improved
A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians
We investigate large-time asymptotics for viscous Hamilton--Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems
Stability of the Steiner symmetrization of convex sets
The isoperimetric inequality for Steiner symmetrization of any
codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets
Rigidity for perimeter inequalities under symmetrization: state of the art and open problems
We review some classical results in symmetrization theory, some recent progress in understanding rigidity, and indicate some open problems
Adjoint methods for obstacle problems and weakly coupled systems of PDE
The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton--Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived
Aubry-Mather measures in the non convex setting
The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053–1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301–319], is used to construct analogues to the Aubry–Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.
Copyright © 2011 Society for Industrial and Applied Mathematic
Essential connectedness and the rigidity problem for Gaussian symmetrization
We provide a geometric characterization of rigidity of equality cases in
Ehrhard's symmetrization inequality for Gaussian perimeter. This condition is
formulated in terms of a new measure-theoretic notion of connectedness for
Borel sets, inspired by Federer's definition of indecomposable current.Comment: 38 page
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