We study the integral representation properties of limits of sequences ofintegral functionals like   $\int f(x,Du)\,{\rm d}x$   undernonstandard growth conditions of (p,q)-type: namely, we assume that $$\vert z\vert^{p(x)}\leq f(x,z)\leq L(1+\vert z\vert^{p(x)})\,.$$ Under weak assumptions on the continuous function p(x), we proveΓ-convergence to integral functionals of the same type.We also analyse the case of integrands f(x,u,Du) depending explicitlyon u; finally we weaken the assumption allowing p(x) to bediscontinuous on nice sets.

We prove a logarithmic stability estimate fora parabolic inverse problem concerning the localization of unknowncavities in a thermicconducting medium Ω in ${\mathbb R}^n$ , n ≥ 2, from a singlepair of boundarymeasurements of temperature and thermal flux.

We study the stability of a flexible beam clamped at one end. Amass is attached at the other end, where a control moment isapplied. The boundary control is proportional to the angular velocityat the end. By spectral analysis, we prove that the optimal decay rateof the energy is given by the spectrum of the generator of thesemigroup associated to the system.

In this paper we establish the existence of a positive solutionfor an asymptotically linear elliptic problem on $\xR^N$ . The maindifficulties to overcome are the lack of a priori bounds forPalais–Smale sequences and a lack of compactness as the domain isunbounded. For the first one we make use of techniques introducedby Lions in his work on concentration compactness. For thesecond we show how the fact that the “Problem at infinity” isautonomous, in contrast to just periodic, can be used in order toregain compactness.