Mathematics

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely ?�div ???????�u 1?�|?�u | 2 ????????????????????????=?in R N , where N?? . We first prove a new gradient estimate for classical solutions with smooth data ? . As a consequence we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class W 2,q and strictly spacelike in R N , provided that ???L q ( R N )??L m ( R N ) with q>N and m?�[1, 2N N+2 ] .

We adapt the argument of Dodson-Murphy to give a simple proof of scattering below the ground state for the intercritical inhomogeneous nonlinear Schrödinger equation. The decaying factor in the nonlinearity obviates the need for a radial assumption.

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation ?�Δu+u=a(x)|u | p?? u in an annulus A??R N ( N?? ). Here p>2 is allowed to be supercritical and a(x) is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution u we construct. In the case where a equals a positive constant, we obtain nonradial solutions in the case where the exponent p is large or when the annulus A is large with fixed width.

This manuscript focus on in the transmission problem for one dimensional waves with nonlinear weights on the frictional damping and time-varying delay. We prove global existence of solutions using Kato's variable norm technique and we show the exponential stability by the energy method with the construction of a suitable Lyapunov functional.

In this note we prove that the sine-Gordon breather is the only quasimonochromatic breather in the context of nonlinear wave equations in R N .

We use profile decomposition to characterize 2-soliton solutions of the KdV equation as global minimizers to a constrained variational problem involving three of the polynomial conservation laws for the KdV equation.

The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. We show an analog of the Lanford's theorem in this domain, with specular reflection boundary condition, stating the convergence in the low density limit of the first marginal of the density function of a system of N hard spheres towards the solution of the Boltzmann equation associated to the initial data corresponding to the initial state of the one-particle-density function. The original contributions of this work consist in two main points: the rigorous definition of the collision operator and of the functional space in which the BBGKY hierarchy is solved in a strong sense; and the adaptation to the case of the half-space of the control of the recollisions performed by Gallagher, Saint-Raymond and Texier, which is a crucial step to obtain the Lanford's theorem.

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

We present a three dimensional, time dependent model for bone regeneration in the presence of porous scaffolds to bridge critical size bone defects. Our approach uses homogenized quantities, thus drastically reducing computational cost compared to models resolving the microstructural scale of the scaffold. Using abstract functional relationships instead of concrete effective material properties, our model can incorporate the homogenized material tensors for a large class of scaffold microstructure designs. We prove an existence and uniqueness theorem for solutions based on a fixed point argument. We include the cases of mixed boundary conditions and multiple, interacting signalling molecules, both being important for application. Furthermore we present numerical simulations showing good agreement with experimental findings.

We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical p -Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional p -Laplacian problems.