Maximum Principles for k-Hessian Equations with Lower Order Terms on Unbounded Domains

Bhattacharya, Tilak; Mohammed, Ahmed

doi:10.1007/s12220-020-00415-0

Cited by 15 publications

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“…a sharp theorem for the variable exponent p-Laplace equation, and also sharp results for equations allowing for sublinear growth in the gradient. Phragmén-Lindelöf theorems for plurisubharmonic functions on cones were proved in [52] while k-Hessian equations with lower order terms were considered in [15]. The present paper complements the above by giving the sharp exponent k(ν, p) explicitly in case of positive p-harmonic functions in planar sectors.…”

Section: Introductionmentioning

confidence: 70%

Estimates of $p$-harmonic functions in planar sectors

Lundström

Singh

2023

Arkiv För Matematik

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Suppose that p∈(1, ∞], ν ∈[1/2, ∞), Sν = (x 1 , x 2 )∈R 2 \{(0, 0)}:|φ|< π 2ν , where φ is the polar angle of (x 1 , x 2 ). 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 thatwhenever x∈B(0, R)∩S 2ν 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 R n . Finally, when ν ∈(1/2, ∞) and p∈(1, ∞) we prove uniqueness (modulo normalization) of positive p-harmonic functions in Sν vanishing on ∂Sν .

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Section: Introductionmentioning

confidence: 70%