关于凸区域上椭圆方程的弱解及其导数的L^2-估计

The L^2 estimates for the weak solution and its derivatives to elliptic equations on the convex domains

研究了有界凸区域上非齐次散度型二阶椭圆方程- i( aij( X) ju( X) +bi( X) u( X) ) +ci( X) iu( X) +c( X) u( X) =f( X)的非零 H1 0 -解 u的二阶导数的 L2估计 ,得到弱解 u∈H2 ( D)∩H1 0 ( D)且有精确估计‖u‖H2 (D) ≤Cdiam(D) 2 ·‖ f‖ L2 ( D) ,常数 C与区域 D的直径等无关 .

The L 2 resolvent estimates are discussed for the weak solution and its derivatives to the following divergence form elliptic equations in convex domains D - i(a ij (X) ju(X)+b i(X)u(x))+c i(X) iu(X)+c(X)u(X)=f(X) And some optimal estimates for the weak solution u∈H 2(D)∩H 0(D)are giren as follows ‖u‖ H 2(D) ≤C diam(D) 2·‖f‖ L 2(D) Where the constant C is independent of the diameter of the domain D.