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.

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.

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.

It is shown that the punctual quotient scheme$Q$_{l}^{r}parametrizing all zero-dimensional quotients $$\mathcal{O}_{A^2 }^{ \oplus ^r } \to T$$ of length$l$and supported at some fixed point O∈$A$^{2}in the plane is irreducible.

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