LOCAL BOUNDEDNESS OF NONAUTONOMOUS SUPERPOSITION OPERATORS IN

The main goal of this paper is to give the answer to one of the main problems of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we prove that if the superposition operator maps the space $BV[0,1]$ into itself, then it is automatically locally bounded, provided its generator is a locally bounded function.

Nonexistence theorems for weak solutions of quasilinear elliptic equations

New nonexistence results are obtained for entire bounded (either from above or from below) weak solutions of wide classes of quasilinear elliptic equations and inequalities. It should be stressed that these solutions belong only locally to the corresponding Sobolev spaces. Important examples of the situations considered herein are the following:Σi=1n(a (x)|∇u| p−2uxi)=−|u| q−1u,Σi=1n(a (x)| uxi | p−2uxi)xi=−|u| q−1u,Σi=1n(a (x)|∇u| p−2uxi/1+|∇u| 2)xi=−|u| q−1u, wheren≥1, p>1, q>0are fixed real numbers, anda(x)is a nonnegative measurable locally bounded function. The methods involve the use of capacity theory in connection with special types of test functions and new integral inequalities. Various results, involving mainly classical solutions, are improved and/or extended to the present cases.

Some Remarks on Non-Newtonian Fluids Including Nonconvex Perturbations of the Bingham and Powell–Eyring Model for Viscoplastic Fluids

We consider quasi-static flows of certain viscoplastic materials for which the velocity field v can be found as a minimizer of the functional [Formula: see text] in classes of functions u : ℝn ⊃ Ω → ℝn satisfying div u = 0 and also the appropriate boundary conditions. The density ω is characteristic for the material under consideration and ℰv denotes the symmetric gradient of v. In case of a Bingham fluid we have for example ω(ℰv) = η|ℰv|2 + g|ℰv| with positive constants η and g. We also consider various perturbations of ω which are not assumed to be convex so that we have to study the relaxed variational problem. Our main result states that in all cases the symmetric derivative of the velocity field is a locally bounded function.