In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N...In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N)_(∞)u+β|D u|,which arises from the random game named biased tug-of-war.By studying directly theβ−biased infinity Laplacian equation,we construct the appropriate exponential cones as barrier functions to establish a key estimate.Based on this estimate,we obtain the Harnack inequality,Hopf boundary point lemma,Lipschitz estimate and the Liouville property etc.展开更多
基金the Fundamental Research Funds for the Central Universities(Grant No.30919013235)National Natural Science Foundation of China(Nos.11501292 and 11501293).
文摘In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N)_(∞)u+β|D u|,which arises from the random game named biased tug-of-war.By studying directly theβ−biased infinity Laplacian equation,we construct the appropriate exponential cones as barrier functions to establish a key estimate.Based on this estimate,we obtain the Harnack inequality,Hopf boundary point lemma,Lipschitz estimate and the Liouville property etc.
