The Weighted Estimates On The Solution Of Gellerstedt Equation In The Upper Half Space

Partial differential equations,as an important part of contemporary mathematics,are an important bridge between many branches of pure mathematics and the fields of natural sciences and engineering technology.Gellerstedt equation is an important partial differential equation.This thesis mainly studies the local basic solution of the Gellerstedt operator,the Green function of the Gellerstedt operator in the upper half plane,and the weighted estimate of the second derivative of the solution to the Dirichlet boundary value problem are presented.Firstly,consider the Gellerstedt operator G=t_lΔ_x+（（?）²）/（（?）t²）,t>0,x=(x₁,...,x_n-1),l>0,the equation is degenerate on the line t=0.Assume that t₀>0,x₀∈R^n-1.Transform the domain U₀ near(x₀,t₀ to the domain V₀ near（y₀,1）,y₀∈Rⁿ and transform the operator G to a uniformly elliptic operator with analytic coefficients on V₀.Applying F.John’s results on uniformly elliptic operator with analytic coefficients,we obtain local fundamental solution of the operator G on the domain U₀.Then,on the upper plane R₊²={（x,y）∈R²,y>0}的 the Green function of the Gellerstedt operator G=y^l（（?）²）/（（?）x²）+（（?）²）/（（?）y²） is obtained by hypergeometric function.The Green function is the fundamental solution of G and vanish on the boundary of R₊².Finally,study the Dirichlet value problem the of the Gellerstedt equation on the upper plane:（?）Weighted estimates on the solution of he Dirichlet value problem is obtained.