Caccioppoli inequality for truncated subsolutions

Given: Countable Choice and the Axiom of Choice; an open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n\ge 2$; constants $0<\theta \le M_a^2$; a measurable symmetric coefficient field $A=(a^{ij})$ with $\theta \mathrm{\mid }\xi {\mathrm{\mid }}^2\le ⟨A\xi ,\xi ⟩\le M_a^2\mathrm{\mid }\xi {\mathrm{\mid }}^2$ a.e.; $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$; and a real class $u\in H^1(\mathrm{\Omega };\mathbb{R})$ with $a_0(u,\phi )\le {\int }_{\mathrm{\Omega }}f\phi$ for every nonnegative $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$.

Assume Countable Choice and the Axiom of Choice. For $k\in \mathbb{R}$ and $u\in H^1(\mathrm{\Omega };\mathbb{R})$, the class $u_k:=(u-k{)}^{+}$ lies in $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega };\mathbb{R})$ with $D{u}_k=1_{\{u>k\}}Du$, and $D{u}_k=0$ a.e. on $\{u\le k\}$. Global $H^1(\mathrm{\Omega })$ membership is not asserted for arbitrary $k$ on an infinite-measure domain. For $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ the product ${\eta }^2{u}_k$ lies in $H_0^1(\mathrm{\Omega })$, is nonnegative, and satisfies $D({\eta }^2{u}_k)=2\eta u_{k}D\eta +{\eta }^{2}D{u}_k$ a.e. (Positive-part truncation calculus and admissible cut-off weak tests, Weak subsolutions and supersolutions of a divergence-form equation).

Assume Countable Choice. Every element of $H^1(\mathrm{\Omega })$ has weak first derivatives in $L^2(\mathrm{\Omega })$, the weak derivative is linear, and products of $L^2$ classes with bounded measurable coefficients are integrable on compact sets (Integer-order Sobolev spaces and their norms).

Bumps: for $0<r<R$ there is $\eta \in C_c^{\mathrm{\infty }}(B_R(x_0))$ with $0\le \eta \le 1$, $\eta =1$ on $B_r(x_0)$ and $\mathrm{\mid }D\eta \mathrm{\mid }\le C_U\mathrm{/}(R-r)$ for a universal constant $C_U$; the explicit radial bump $\eta (x)=\sigma ((s^2-\mathrm{\mid }x-x_0{\mathrm{\mid }}^2)\mathrm{/}(s^2-r^2))$, $s=(r+R)\mathrm{/}2$, of A smooth bump between concentric Euclidean balls and Compactly supported scaled Euclidean bumps provides it, since on the support $\mathrm{\mid }x-x_0\mathrm{\mid }\le s$ and the chain rule give $\mathrm{\mid }D\eta \mathrm{\mid }\le \mathrm{\parallel }{\sigma }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}2s\mathrm{/}(s^2-r^2)\le \mathrm{\parallel }{\sigma }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}4\mathrm{/}(R-r)$.

Young's inequality with conjugate exponents $p=q=2$ and weight: for $a,b\ge 0$ and $\epsilon >0$, $2ab\le \epsilon a^2+b^2\mathrm{/}\epsilon$ (Young's inequality for conjugate real exponents); Cauchy–Schwarz in $L^2$ gives $\mathrm{\mid }\int gh\mathrm{\mid }\le (\int g^2{)}^{1\mathrm{/}2}(\int h^2{)}^{1\mathrm{/}2}$ (Holder's inequality for integrals, including the endpoint cases).