Harnack inequality for nonnegative weak solutions

Given: Countable Choice and the Axiom of Choice; an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n\ge 2$; uniformly elliptic measurable symmetric coefficients $A$ with constants $\theta ,M_a$; the principal operator $L_{0}u=-D_i(a^{ij}{D}_{j}u)$ with form $a_0$; a source $F\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^q(\mathrm{\Omega })$, $q>n\mathrm{/}2$; a nonnegative $u\in H^1(\mathrm{\Omega };\mathbb{R})$ with $a_0(u,\phi )=-{\int }_{\mathrm{\Omega }}F\phi dx$ for every real $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$; a ball $B_R(x_0)$ with $B_{2R}(x_0)⋐\mathrm{\Omega }$.

Both roles of a local solution: the identity $a_0(u,\phi )=-{\int }_{\mathrm{\Omega }}F\phi$ against real compactly supported smooth tests gives both the subsolution inequality and the supersolution inequality for the equation $L_{0}u=-F$, with the appropriate inequality directions (Weak subsolutions and supersolutions of a divergence-form equation, Uniformly elliptic divergence-form operators and their sesquilinear forms).

Assume the Axiom of Choice. Local boundedness with a scale-correct source: for every ball $B_S(y)⋐\mathrm{\Omega }$, every $0<\rho <1$ and every $p>0$, ${\mathrm{ess\hspace{0.17em}sup}}_{B_{\rho S}(y)}w\le C_1[(\frac{1}{\mathrm{\mid }{B}_S(y)\mathrm{\mid }}{\int }_{B_S(y)}{w}^p{)}^{1\mathrm{/}p}+S^{2-n\mathrm{/}q}\mathrm{\parallel }{f}^{+}{\mathrm{\parallel }}_{L^q(B_S(y))}]$ for every nonnegative weak subsolution $w$ of $a_0(w,\cdot )\le \int f\cdot dx$ with $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^q(\mathrm{\Omega })$, where $C_1=C_1(n,q,\theta ,M_a,\rho ,p)$ (De Giorgi local boundedness with a scale-correct forcing term).

Assume the Axiom of Choice. Weak Harnack inequality: for every ball $B_S(y)$ with $B_{2S}(y)⋐\mathrm{\Omega }$ and every $0<p<n\mathrm{/}(n-2)$ when $n\ge 3$, or every finite $p>0$ when $n=2$, $S^{-n\mathrm{/}p}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^p(B_S(y))}\le C_2({\mathrm{ess\hspace{0.17em}inf}}_{B_{S\mathrm{/}2}(y)}w+S^{2-n\mathrm{/}q}\mathrm{\parallel }G{\mathrm{\parallel }}_{L^q(B_{2S}(y))})$ for every nonnegative weak supersolution $w$ of $L_{0}w=-G$ with $G\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^q(\mathrm{\Omega })$, where $C_2=C_2(n,q,\theta ,M_a,p)$; the range contains $s_0:=\min \{p_0,1\mathrm{/}2\}$, where $p_0=p_0(n,\theta ,M_a)>0$ is produced by the Moser iteration (Weak Harnack inequality for nonnegative supersolutions, Moser iteration for positive supersolutions: negative-power and logarithmic comparison).

Assume the Axiom of Choice. Averaging and the elementary comparison of the negative part of the source: $\frac{1}{\mathrm{\mid }{B}_S(y)\mathrm{\mid }}{\int }_{B_S(y)}{w}^{p}dx=S^{-n}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^p(B_S(y))}^p\mathrm{/}\mathrm{\mid }{B}_1\mathrm{\mid }$, and $\mathrm{\parallel }(-F{)}^{+}{\mathrm{\parallel }}_{L^q}=\mathrm{\parallel }{F}^{-}{\mathrm{\parallel }}_{L^q}\le \mathrm{\parallel }F{\mathrm{\parallel }}_{L^q}$ (The average of a locally integrable function over a Euclidean ball, The space $L^p(\mu )$ as the quotient by null functions, The essential supremum of a measurable function with respect to a measure).

Assume the Axiom of Choice. The $n=2$ substitute: the critical embedding $W_0^{1,2}\hookrightarrow L^{\kappa }$ for every finite $\kappa$ replaces the $2^{\ast }$ embedding in both quoted theorems (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^q$).