Research Of Convexity For P-harmonic Functions

Convexity has always been a hot topic in elliptic partial differential equations,so it is necessary to study convexity.One of the important ways to prove the relevant convexity of a solution u is to obtain certain convexity estimates of the solution,which is to construct a suitable auxiliary function ψ,and prove that ψsatisfies the differential inequality,that is to say,it is greater than or equal to zero for the elliptic operator acting on the auxiliary function.Then using the minimum principle we prove that its maximum value is obtained on the boundary,and then get the convexity estimate of the solution.In this paper,we mainly study the convexity estimate of solutions to the following elliptic equation Dirichlet problem in two-dimensional case where p>2.The solution to this problem is called the p-Green’s function of Ωwith pole at x0,where Ω is a smooth,bounded and convex domain in R2.Firstly,let v＝up-1/p-2 and we construct a suitable auxiliary function ψ=vp-4/p-1 det D2v;Secondly,we prove that the auxiliary function satisfies the differential inequality aαβψαβ≤0 mod(Dψ),where aαβ=|Du|2δαβ+(p-2)uαuβ.Finally,with the minimum principle,we can obtain that the function attains its minimum on the boundary.Furthermore,we can obtain the convexity estimate of the solution.