Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).

We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann map$f$is topologically conjugate to a holomorphic map$g$, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then$g$is Collet-Eckmann, too.

A solution to a Dirichlet problem for the complex Monge-Ampère operator is characterized as a minimizer of an energy functional. A mutual energy estimate and a generalization of Hölder's inequality is proved. A comparison is made with corresponding results in classical potential theory.

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=−1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections ($N$∩$L$_{p}(w_{0}), N∩L_{p}(w_{1})), where$N$is the linear space which consists of all functions with the integral equal to 0. We calculate the$K$-functional for the couple ($N$∩$L$_{p}(w_{0}),$N$∩$L$_{p}(w_{1})), which turns out to be essentially different from the$K$-functional for ($L$_{p}(w_{0}), L_{p}(w_{1})), even for the case when$N$∩$L$_{p}(w_{i}) is dense in$L$_{p}(w_{i}) ($i$=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.

The equation$x$^{(n)}(t)=(−1)^{$n$}│$x(t)$│^{$k$}with$k$>1 is considered. In the case$n$≦4 it is proved that solutions defined in a neighbourhood of infinity coincide with$C$(t−t_{0})^{−n/(k−1)}, where$C$is a constant depending only on$n$and$k$. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times ($t−t$_{0})^{−n/(k−1)}. It is shown that they do not necessarily coincide with$C$(t−t_{0})^{−n/(k−1)}. This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.