De Giorgi local boundedness with a scale-correct forcing term

Given: Countable Choice and the Axiom of Choice; an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n\ge 2$; constants $0<\theta \le M_a^2$; measurable symmetric coefficients $A$ with $\theta \mathrm{\mid }\xi {\mathrm{\mid }}^2\le ⟨A(x)\xi ,\xi ⟩\le M_a^2\mathrm{\mid }\xi {\mathrm{\mid }}^2$; 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 $a_0(u,\phi )\le {\int }_{\mathrm{\Omega }}f\phi dx$ for all nonnegative $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$; a ball $B_R(x_0)⋐\mathrm{\Omega }$ and $0<\rho <1$.

Assume Countable Choice and the Axiom of Choice. Sobolev positive-part calculus: for $w\in H^1(\mathrm{\Omega };\mathbb{R})$, $w^{+}\in H^1$ with $D{w}^{+}={\mathbf{1}}_{\{w>0\}}Dw$ and $Dw=0$ a.e. on $\{w=0\}$; multiplication by a compactly supported smooth factor obeys the weak product rule. If $a_0(w,v)\le 0$ for every nonnegative $v\in H_0^1(\mathrm{\Omega })$, then $w^{+}$ is a weak subsolution of the principal operator, so $a_0(w^{+},v)\le 0$ for every nonnegative $v\in H_0^1(\mathrm{\Omega })$. Here is the admissible truncation proof. For $\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, $\varphi \ge 0$, set ${\eta }_{ϵ}(t)=\min \{1,t^{+}\mathrm{/}ϵ\}$. The chain and product rules give ${\psi }_{ϵ}=\varphi {\eta }_{ϵ}(w)\in H^1$ with compact support in $\mathrm{\Omega }$, hence ${\psi }_{ϵ}\in H_0^1(\mathrm{\Omega })$; it is nonnegative, and $D_i{\psi }_{ϵ}={\eta }_{ϵ}(w)D_i\varphi +{ϵ}^{-1}\varphi {\mathbf{1}}_{\{0<w<ϵ\}}{D}_{i}w$ (the level-set endpoints contribute zero because Sobolev gradients vanish a.e. on a level set). Thus $a_0(w,{\psi }_{ϵ})={\int }_{\mathrm{\Omega }}{\eta }_{ϵ}(w)a^{ij}{D}_{j}w{D}_i\varphi dx+\frac{1}{ϵ}{\int }_{\{0<w<ϵ\}}\varphi a^{ij}{D}_{j}w{D}_{i}wdx\le 0.$ The second integral is nonnegative by symmetry, ellipticity, and $\varphi \ge 0$, so the first is nonpositive. As $ϵ\downarrow 0$, ${\eta }_{ϵ}(w)\to {\mathbf{1}}_{\{w>0\}}$ pointwise and is bounded by $1$; dominated convergence applies because $A$ is bounded and $\mathrm{\mid }Dw\mathrm{\mid }\mathrm{\mid }D\varphi \mathrm{\mid }$ is integrable on $\mathrm{supp}\varphi$. Using $D{w}^{+}={\mathbf{1}}_{\{w>0\}}Dw$ gives $a_0(w^{+},\varphi )\le 0$. To extend from smooth tests to every nonnegative $v\in H_0^1$, choose $z_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $z_j\to v$ in $H^1$ and pass to a subsequence with $z_j\to v$ a.e. The positive-part gradient formulas give $D{z}_j^{+}-Dv={\mathbf{1}}_{\{z_j>0\}}(D{z}_j-Dv)+({\mathbf{1}}_{\{z_j>0\}}-{\mathbf{1}}_{\{v>0\}})Dv\mathrm{.}$ The first term tends to zero in $L^2$; the second does too by dominated convergence, since its indicator tends to zero on $\{v>0\}$ and $Dv=0$ a.e. on $\{v=0\}$. Also $z_j^{+}\to v$ in $L^2$ by the $1$-Lipschitz property, hence $z_j^{+}\to v$ in $H^1$. Each nonzero $z_j^{+}$ has compact support in $\mathrm{\Omega }$; zero-extend it and convolve with a nonnegative unit-mass radial mollifier, chosen with support radius smaller than $\mathrm{dist}(\mathrm{supp}{z}_j^{+},\mathrm{\partial }\mathrm{\Omega })$ (if $z_j^{+}=0$, keep the zero function). The mollified functions are nonnegative and in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, and converge to $z_j^{+}$ in $H^1$ by approximate-identity convergence applied to the function and its weak gradient. A diagonal choice gives nonnegative smooth tests converging to $v$. Boundedness of $a_0(w,\cdot )$ passes the inequality to $v$. In particular, no product of an indicator with an arbitrary test is asserted to lie in $H_0^1$. (Positive-part truncation calculus and admissible cut-off weak tests, Positive, negative, and truncated Sobolev functions, Chain rule for globally Lipschitz scalar maps of Sobolev functions, Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order, A radial mollifier family in Rn, A smooth bump between concentric Euclidean balls, The mollifier family generated by a unit-mass smooth bump, Interior mollification commutes with weak derivatives, Every $L^1$ approximate identity converges to the identity in $L^p$ for $1\le p<\mathrm{\infty }$, Dominated convergence, Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms, The elliptic form is well defined and bounded on $H^1$, Uniformly elliptic divergence-form operators and their sesquilinear forms, Weak subsolutions and supersolutions of a divergence-form equation).

Assume the Axiom of Choice. Lax-Milgram and coercivity: $H^1(B_1)$ is Hilbert by $H^k$ is a Hilbert space under the derivative-sum inner product. Its subspace $H_0^1(B_1)$ is closed and linear by its closure definition: a Cauchy sequence there converges in $H^1(B_1)$, and its limit still lies in the closure of the smooth tests. Thus the restricted derivative-sum inner product makes $H_0^1(B_1)$ Hilbert. The form $a_0$ is a bounded sesquilinear form on the Hilbert space $H_0^1(B_1)$ with coercivity constant $\alpha =\theta \mathrm{/}(1+C_P(B_1{)}^2)$, and for every bounded conjugate-linear functional $G$ on $H_0^1(B_1)$ there is a unique $h\in H_0^1(B_1)$ with $a_0(h,v)=G(v)$ for all $v\in H_0^1(B_1)$, the weak Dirichlet solution; it satisfies $\alpha \mathrm{\parallel }h{\mathrm{\parallel }}_{H_0^1}\le \mathrm{\parallel }G{\mathrm{\parallel }}_{H^{-1}}$ (The Lax--Milgram theorem, Coercivity of the principal Dirichlet form, The elliptic form is well defined and bounded on $H^1$, The negative Sobolev space $H^{-1}(\mathrm{\Omega })$, Weak Dirichlet solutions for a divergence-form operator, Zero-boundary Sobolev space as a norm closure).

Assume the Axiom of Choice. Embedding of $H_0^1$ into the dual exponents of $L^q$: for $n\ge 3$ and $1\le r<2^{\ast }=2n\mathrm{/}(n-2)$ there is $C_r$ with $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^r(B_1)}\le C_r\mathrm{\parallel }w{\mathrm{\parallel }}_{H_0^1(B_1)}$ for all $w\in H_0^1(B_1)$, by Holder and the Sobolev inequality; for $n=2$ the same holds for every finite $r$ by the critical embedding $W_0^{1,2}(B_1)\hookrightarrow L^r(B_1)$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^q$, Holder's inequality for integrals, including the endpoint cases, The space $L^p(\mu )$ as the quotient by null functions).

Assume the Axiom of Choice. The weak maximum principle with a source on the bounded $C^1$ domain $B_1$: if $v\in H^1(B_1;\mathbb{R})$ satisfies $a_0(v,\phi )\le {\int }_{B_1}g\phi dx$ for every nonnegative $\phi \in H_0^1(B_1)$ with $g\in L^q(B_1)$, $q>n\mathrm{/}2$, then ${\mathrm{ess\hspace{0.17em}sup}}_{B_1}v\le {\sup }_{\mathrm{\partial }{B}_1}{v}^{+}+C_0\mathrm{\parallel }{g}^{+}{\mathrm{\parallel }}_{L^q(B_1)}$ with $C_0=C_0(n,q,\theta ,M_a,B_1)$; in particular for $v\in H_0^1(B_1)$ one has ${\mathrm{ess\hspace{0.17em}sup}}_{B_1}v\le C_0\mathrm{\parallel }{g}^{+}{\mathrm{\parallel }}_{L^q(B_1)}$, and if $g\le 0$ then $v\le 0$ a.e. (Weak maximum principle for coercive divergence-form equations, The $L^p$ trace operator on a bounded $C^1$ domain, The kernel of the trace is the closure of the test functions, Bounded C^k domains and boundary charts).

Assume the Axiom of Choice. Homogeneous local boundedness: if $w\in H^1(B_1;\mathbb{R})$ satisfies $w\ge 0$ a.e. and $a_0(w,v)\le 0$ for every nonnegative $v\in H_0^1(B_1)$, then for every $0<s<1$, every $0<\sigma <1$ and every $p>0$, ${\mathrm{ess\hspace{0.17em}sup}}_{B_{\sigma s}}w\le C_1(\frac{1}{\mathrm{\mid }{B}_s\mathrm{\mid }}{\int }_{B_s}{w}^{p}dx{)}^{1\mathrm{/}p}$; this is the homogeneous theorem applied on the compactly contained ball $B_s⋐B_1$ (De Giorgi local boundedness of homogeneous subsolutions).

Assume the Axiom of Choice. Poincare inequality on $W_0^{1,2}$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, The essential supremum of a measurable function with respect to a measure).