Let be an arbitrary non-empty bounded Lipschitz domain in RM RN. Given > 0, squeeze by the factor in the y-direction to obtain the squeezed domain := {(x, y) | (x, y) }. Let and be positive constants. Consider the following semilinear damped wave equation on , where is the exterior normal vector field on and G is an appropriate nonlinearity, which ensures that (W) generates a (local) flow ̃ on X := H1() L2(). We show that there is a closed subspace X0 of X and a flow ̃0 on X0 that is the limit flow of the family ̃, > 0. We show that, as 0, the family ̃ converges in some singular sense to ̃ and establish a technical singular asymptotic compactness property. As a corollary, we obtain an upper-semicontinuity result for global attractors of the family ̃, 0, generalizing results obtained previously by Hale and Raugel for domains that are ordinate sets of a positive function.

The results obtained here are also applied in our paper On a general Conley index continuation principle for singular perturbation problems to establish a singular Conley index continuation principle for damped wave equations on thin domains.

It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the classical estimates fail. The dependence on the data is explicit. In particular, this provides good control over the domain dependence, which is important for applications involving varying domains.

We study fine properties of currents in the framework of geometric measure theory on metric spaces developed by Ambrosio and Kirchheim, and we prove a rectifiability criterion for flat currents of finite mass. We apply these tools to study the structure of the distributional Jacobians of functions in the space BnV, defined by Jerrard and Soner. We define the subspace of special functions of bounded higher variation and we prove a closure theorem.

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x| , where n R. There are two constants of the motion that affect the large-time behaviour: Hopf considered the case n = 0 (i.e. u0L1(R)), and the case sufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only on and n, such that uniformly in . When n 1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

Let be the open unit disc in C. It is proved that a holomorphic embedding f : C2 can grow arbitrarily fast near b. It is also proved that a holomorphic embedding f : C C2 can grow arbitrarily fast near infinity.

We consider the eigenvalue problem in an arbitrary OrliczSobolev space. We show that the existence of an eigenvalue can be derived from a generalized version of Lagrange multiplier rule. Our approach also applies to more general problems. We emphasize that no 2 condition is imposed.

We consider the behaviour of buckling-driven thin-film blisterings using Von Karman's plate theory. Our results are asymptotic in the thinness and emphasize the incorporation of in-plane displacements in the model, a factor often ignored in the literature. Our work indicates that the inclusion of these displacements has a profound effect on the nature of solutions. We compare different constraints on displacements by estimating the leading order of corresponding energies. Our results strongly suggest that branching and other fine-scale structures arise in thin-film blistering, as has been observed in experiments.

Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in M R+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T 0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem.

More specifically, we show the following.

1. (i) It is possible to determine Rt for any t 0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.

2. (ii) It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.

We consider two model reaction-diffusion systems of bistable type arising in the theory of phase transition; they appear in various physical contexts, such as thin magnetic films and diblock copolymers. We prove the convergence of the solution of these systems to the solution of free-boundary problems involving modified motion by mean curvature.

We introduce the concept of flat ridges for submanifolds of codimension 2 from the viewpoint of their contact with hyperplanes. We characterize them geometrically, studying some of their properties. In particular, we see that the highest-order flat ridges coincide with the flattenings of the asymptotic lines and from this we obtain some lower bounds for their numbers under appropriate conditions.

Radial deformations of a ball composed of a nonlinear elastic material and corresponding to cavitation have been much studied. In this paper we use rescalings to show that each such deformation can be used to construct infinitely many non-symmetric singular weak solutions of the equations of nonlinear elasticity for the same displacement boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date.

We study some asymptotic behaviour of phase interfaces with variable chemical potential under the uniform energy bound. The problem is motivated by the Cahn-Hilliard equation, where one has a control of the total energy and chemical potential. We show that the limit interface is an integral varifold with generalized Lp mean curvature. The convergence of interfaces as 0 is in the Hausdorff distance sense.