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Imported unit-ball Harnack inequality with bounded potential

Statement

Let $n \geq 2$ . On $B_{1} \subset R^{n}$ , let $A$ be real measurable symmetric with $ξ^{T} A ξ \geq ∣ ξ ∣^{2}$ and $\sum_{i, j} ∣ A^{i j} ∣ \leq Λ$ , and let $Q_{0}$ be real measurable with $0 \leq Q_{0} \leq Q$ . A nonnegative weak solution $v \in W^{1, 2} (B_{1})$ of $- div (A \nabla v) + Q_{0} v = 0$ satisfies ${ess sup}_{B_{1 / 2}} v \leq C {ess inf}_{B_{1 / 2}} v$ , with $C$ depending only on $n, Λ, Q$ .

Facts & Assumptions

Given: Work in ZFC, including The Axiom of Choice; this includes the countable choice assumed by the formulation suppliers below. The open ball is Euclidean.

[F1]

Use $W^{1, 2} (B_{1}; R)$ as the space of almost-everywhere classes with square-integrable weak first derivatives from Integer-order Sobolev spaces and their norms. The gradient $\nabla v$ consists of the weak derivatives in Weak derivative of a locally integrable function; $v \geq 0$ means almost everywhere.

[F2]

All null sets and integrals use Lebesgue measure $λ_{n}$ from Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ. Measurability of each real coefficient on $B_{1}$ means that its extension by zero to $R^{n}$ is Lebesgue measurable in the sense of Borel measurable and Lebesgue measurable functions on ℝⁿ. Coefficient bounds are understood almost everywhere.

[F3]

A weak solution means $\int_{B_{1}} (A \nabla v \cdot \nabla φ + Q_{0} v φ) d λ_{n} = 0$ for every real smooth function $φ$ compactly supported in $B_{1}$ . For measurable $E \subseteq B_{1}$ , set ${ess sup}_{E} v = \inf {a \in R : λ_{n} ({x \in E : v (x) > a}) = 0}$ and ${ess inf}_{E} v = \sup {a \in R : λ_{n} ({x \in E : v (x) < a}) = 0}$ , with extended-real infimum and supremum conventions. These expressions ignore changes on null sets.

[F4]

The independent Simon reference supplies the analytic Harnack conclusion with the coefficient and weak-solution hypotheses in External input. It is imported, rather than proved locally.

External input

Depends on

Used by · 1

Multiplicative interior Harnack for geometric Schrödinger solutions

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Leon Simon, Lectures on Partial Differential Equations · Lecture 18, Theorem 1 and its displayed proof, printed p.211 (PDF sheet 108); coefficient hypotheses (i)–(iii), Lecture 17, printed pp.199–200 (PDF sheets 102–103), including the bounded-measurable q=∞ convention and essential-supremum remark on p.200; specialization R=1 and θ=1/2.

Let $n\ge2$. On $B_1\subset\mathbb R^n$, let $A$ be real measurable symmetric with $\xi^TA\xi\ge|\xi|^2$ and $\sum_{i,j}|A^{ij}|\le\Lambda$, and let $Q_0$ be real measurable with $0\le Q_0\le Q$. A nonnegative weak solution $v\in W^{1,2}(B_1)$ of $-\operatorname{div}(A\nabla v)+Q_0v=0$ satisfies $\operatorname{ess\,sup}_{B_{1/2}}v\le C\operatorname{ess\,inf}_{B_{1/2}}v$, with $C$ depending only on $n,\Lambda,Q$.

Hypotheses: Use the source's multi-index operator with a_{e_i,e_j}=A^{ij}, a_{00}=Q_0, all first-order coefficients zero. Its normalized ellipticity is (i), its top coefficient bound is (ii), and on the unit ball condition (iii) holds with q=∞ and Γ=√Q. Theorem 1 uses θ=1/2. Supremum/infimum are understood essentially for weak representatives; for the present continuous solutions these are the ordinary extrema.

Reading limits: For this adjudication, opened the 118-sheet author PDF through the browser and read the extracted text of the coefficient hypotheses (i)–(iii) on printed p.199, the q=∞ and essential-supremum conventions on p.200, and the full Theorem 1 statement and short proof on p.211. The proof invokes Lecture 17 Theorems 1–2; their iteration proofs were not reconstructed or read in full. Only the unit-ball bounded-potential specialization is imported. No general-radius interpretation of the printed lower-order scaling expression is used. No claim of a fresh local PDF download or rendered-image inspection is made.

Sources

Leon Simon, Lectures on Partial Differential Equations · Lecture 18, Theorem 1 and its displayed proof, printed p.211 (PDF sheet 108); coefficient hypotheses (i)–(iii), Lecture 17, printed pp.199–200 (PDF sheets 102–103), including the bounded-measurable q=∞ convention and essential-supremum remark on p.200; specialization R=1 and θ=1/2.