ScalingStacks

Lemma 6.3 . [021S]

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Lemma 6.3.

Suppose u≥0u\geq 0 satisfies in B1⊂ℝdB_{1}\subset\mathbb{R}^{d}:

∂i(ai​j​∂ju)≥f​u+g.\partial_{i}\big(a^{ij}\partial_{j}u\big)\geq fu+g.

Here 1λ⁡(x)≤ai​j​(x)≤λ⁡(x)\frac{1}{\lambda(x)}\leq a^{ij}(x)\leq\lambda(x), with λ⁡(x)∈Lp​(B1)\lambda(x)\in L^{p}(B_{1}), f,g∈Lp/2​(B1)f,\,g\in L^{p/2}(B_{1}) for some p>3​d2p>\frac{3d}{2}, then there exists a constant CC, depending on pp, ‖λ‖Lp​(B1)||\lambda||_{L^{p}(B_{1})}, ‖f‖Lp/2​(B1)||f||_{L^{p/2}(B_{1})}, ‖g‖Lp/2​(B1)||g||_{L^{p/2}(B_{1})}, such that

supB12u≤C⁡(‖u‖L1​(B1)+1).\sup_{B_{\frac{1}{2}}}u\leq C(||u||_{L^{1}(B_{1})}+1).

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