In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $$\Delta u = \lambda_{+\chi_{\{ u>0 \}}} - \lambda_{-\chi_{\{ u<0 \}}},\quad\lambda_{\pm}>0. $$ We prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data$f$and its first and second derivatives vanish at the touch-point.

We show that if the graph of an analytic function in the unit disk$D$is not complete pluripolar in$C$^{2}then the projection of its pluripolar hull contains a fine neighborhood of a point $p\in\partial\mathbf{D}$ . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function$f$in$D$extends to a function ℱ which is defined on a fine neighborhood of a point $p\in\partial\mathbf{D}$ and is finely analytic at$p$then the pluripolar hull of the graph of$f$contains the graph of ℱ over a smaller fine neighborhood of$p$. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them$C$^{∞}functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.

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Auslander–Reiten sequences are the central item of Auslander–Reiten theory, which is one of the most important techniques for the investigation of the structure of abelian categories.

The problem of$subextension$of plurisubharmonic functions is considered. Recently it was shown by Cegrell–Zeriahi, that subextension is always possible for negative plurisubharmonic functions in the energy class ℱ. In this paper we construct, for every hyperconvex domain Ω, a negative plurisubharmonic function in the class ℰ which cannot be subextended. Given any pseudoconvex domain we construct a pluriharmonic function that cannot be subextended.