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Suppose that p∈(1,∞], ν∈[1/2,∞), Sν = (x1, x2)∈R² \{(0, 0)}: |φ|< π/2ν, where φ is the polar angle of (x1, x2). Let R>0 and ωp(x) be the p-harmonic measure of ∂B(0, R)∩Sν at x with respect to B(0, R)∩Sν. We prove that there exists a constant C such that C−1 (|x|/R)k(ν,p) ≤ ωp(x) ≤ C (|x|/R)k(ν,p) whenever x∈B(0, R)∩S2ν and where the exponent k(ν, p) is given explicitly as a function of ν and p. Using this estimate we derive local growth estimates for p-sub- and p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmén-Lindelöf theorem for p-subharmonic functions in the unbounded sector Sν. Moreover, if p=∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in Rn. Finally, when ν∈(1/2, ∞) and p∈(1,∞) we prove uniqueness (modulo normalization) of positive p-harmonic functions in Sν vanishing on ∂Sν. [ABSTRACT FROM AUTHOR]

Phenological changes among plants due to climate change are well documented, but often hard to interpret. In order to assess the adaptive value of observed changes, we study how annual plants with and without growth constraints should optimize their flowering time when productivity and season length changes. We consider growth constraints that depend on the plant's vegetative mass: self-shading, costs for nonphotosynthetic structural tissue and sibling competition., We derive the optimal flowering time from a dynamic energy allocation model using optimal control theory. We prove that an immediate switch (bang-bang control) from vegetative to reproductive growth is optimal with constrained growth and constant mortality., Increasing mean productivity, while keeping season length constant and growth unconstrained, delayed the optimal flowering time. When growth was constrained and productivity was relatively high, the optimal flowering time advanced instead. When the growth season was extended equally at both ends, the optimal flowering time was advanced under constrained growth and delayed under unconstrained growth., Our results suggests that growth constraints are key factors to consider when interpreting phenological flowering responses. It can help to explain phenological patterns along productivity gradients, and links empirical observations made on calendar scales with life-history theory. [ABSTRACT FROM AUTHOR]