MathSciDoc: An Archive for Mathematician ∫

This paper examines the boundary behaviour of superharmonic functions on a half-space in terms of their behaviour along lines normal to the boundary. It is shown that, if the set of lines along which such functions grow quickly is (in a certain sense) metrically dense, then the set of lines along which they are bounded below is topologically small.

We construct singular solutions to equations $$div\mathcal{A}(x,\nabla u) = 0,$$ similar to the$p$-Laplacian, that tend to ∞ on a given closed set of$p$-capacity zero. Moreover, we show that every$G$_{δ}-set of vanishing$p$-capacity is the infinity set of some$A$-superharmonic function.

We study removable sets for the Sobolev space$W$^{1,p}. We show that removability for sets lying in a hyperplane is essentially determined by their thickness measured in terms of a concept of$p$-porosity.

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.

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.

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.

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.

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).

For Ξ∈$R$^{$n$},$t$∈$R$and$f$∈$S$($R$^{$n$}) define $$\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$$ . We determine the optimal regularity$s$_{0}such that $$\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$$ holds where$C$is independent of$f$∈$S$($R$^{$n$}) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11].

Let$M$be a connected, noncompact, complete Riemannian manifold, consider the operator$L=Δ+∇V$for some$V∈C$^{2}(M) with exp[$V$] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of$L$. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_{ϱ→∞}sup$L$_{ϱ<0}while the spectral gap does not exist if o is a pole and lim_{ϱ→∞}inf$L$_{ϱ≥0}. Moreover, the elliptic operators on$R$^{$d$}are also studied.