Pome5D1E62F09217Proposition 8.8.1.5D1E62F09217 Let $a>0$ and let smooth $H_ j$ converge uniformly to a continuous $H$. The smooth solutions of $H_ m(w_ j)=e^{aw_ j+H_ j}\omega ^ n$ converge uniformly to the unique continuous admissible solution of $H_ m(w)=e^{aw+H}\omega ^ n$. For two continuous data $H_1,H_2$, the corresponding solutions satisfy
They also satisfy $-\max _ XH/a\leq w\leq -\min _ XH/a$.
Proof. Smooth solutions exist by 8.6.1. At a maximum of $w_1-w_2$, monotonicity of the smooth Hessian operator gives $e^{aw_1+H_1}\leq e^{aw_2+H_2}$. Hence $a(w_1-w_2)\leq H_2-H_1$ there. Exchanging the functions gives the stated norm estimate for smooth data. Thus a uniformly convergent sequence of smooth data gives a uniformly Cauchy sequence of solutions. Its limit is admissible by the linear distributional inequalities, and 8.7.1 passes the equation to the limit. The bounds follow from those for the smooth solutions. Taking limits in the smooth norm estimate proves it for continuous data. Finally, 8.7.9 in both directions gives uniqueness among continuous admissible solutions. $\square$
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