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De Giorgi oscillation reduction: one half-level set is small

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ be a weak solution of $L_{0} u = 0$ on $Ω$ . Write $M : = {ess sup}_{B_{R} (x_{0})} u$ , $m : = {ess inf}_{B_{R} (x_{0})} u$ and ${osc}_{B_{R} (x_{0})} u : = M - m$ for balls $B_{R} (x_{0}) ⋐ Ω$ . Then the two half-level sets cannot both be large, and one of them is small enough to reduce the oscillation:

(dichotomy) at least one of $∣ {u > (M + m) / 2} \cap B_{R} (x_{0}) ∣$ and $∣ {u < (M + m) / 2} \cap B_{R} (x_{0}) ∣$ is at most $\frac{1}{2} ∣ B_{R} (x_{0}) ∣$ ;
(quantitative reduction) there are constants $η = η (n, θ, M_{a}) \in (0, 1)$ and $C = C (n, θ, M_{a})$ such that for every $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ , ${ess osc}_{B_{R / 2} (x_{0})} u \leq η {ess osc}_{B_{R} (x_{0})} u .$ Moreover the constant $η$ may be chosen as $1 - η_{0} / 2$ where $η_{0} > 0$ depends only on $n, θ, M_{a}$ ; the proof uses localized truncated Caccioppoli estimates and applies the local boundedness estimate De Giorgi local boundedness of homogeneous subsolutions to a nonnegative truncation on the inner ball.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open $Ω \subseteq R^{n}$ , $n \geq 2$ ; a measurable symmetric coefficient field $A$ with $θ ∣ ξ ∣^{2} \leq ⟨ A ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ a.e.; a weak solution $u \in H^{1} (Ω; R)$ of $L_{0} u = 0$ ; and a ball $B_{2 R} (x_{0}) ⋐ Ω$ .

[F1]

Assume Countable Choice and the Axiom of Choice. Local boundedness: every nonnegative weak subsolution $w$ of $L_{0} w = 0$ on an open set satisfies ${ess sup}_{B_{ρ r} (y)} w \leq C_{1} (ρ) (\frac{1}{∣ B_{r} (y) ∣} \int_{B_{r} (y)} w^{2})^{1 / 2}$ for every $B_{r} (y) ⋐ Ω$ and every $0 < ρ < 1$ , with $C_{1} (ρ) = C_{1} (n, θ, M_{a}, ρ)$ (De Giorgi local boundedness of homogeneous subsolutions).

[F2]

Truncated Caccioppoli estimate and truncation subsolution property. For a solution $u$ of $L_{0} u = 0$ and any $k$ , choose smooth nondecreasing $χ_{ϵ}$ with $χ_{ϵ} = 0$ on $(- \infty, 0]$ , $χ_{ϵ} = 1$ on $[ϵ, \infty)$ , and $χ_{ϵ} \geq 0$ . Testing the local equation with $φ χ_{ϵ} (u - k)$ for nonnegative $φ \in C_{c}^{\infty}$ is justified by $H_{0}^{1}$ density; expansion gives $0 = \int χ_{ϵ} (u - k) A D u \cdot D φ + \int φ χ_{ϵ}^{'} (u - k) A D u \cdot D u$ , so the first integral is nonpositive. Letting $ϵ ↓ 0$ , the Sobolev chain rule and $D u = 0$ a.e. on ${u = k}$ give $a_{0} ((u - k)^{+}, φ) \leq 0$ . Thus $(u - k)^{+}$ is a nonnegative local weak subsolution. Also, for $B_{r} ⋐ B_{R}$ , $\int_{B_{r}} ∣ D (u - k)^{+} ∣^{2} \leq C_{0} (R - r)^{- 2} \int_{B_{R}} (u - k)^{+ 2}$ with $C_{0} = C_{0} (θ, M_{a})$ (Local weak solutions of a divergence-form operator, Weak subsolutions and supersolutions of a divergence-form equation, Positive-part truncation calculus and admissible cut-off weak tests, Caccioppoli inequality for truncated subsolutions, Sobolev level-set step: energy decay with explicit level gap and radius loss).

[F3]

Assume the Axiom of Choice. Smooth functions on the closed ball are dense in $H^{1} (B_{R})$ , and $H^{1} (B_{R})$ is the closure of $C^{\infty} ({\overline{B}}_{R})$ under the Sobolev norm; a.e. convergence and $L^{2}$ convergence of the gradients may be assumed along a subsequence (Ambient smooth restrictions are dense on bounded C^k domains).

[F4]

Measure conventions: $ess sup$ and $ess inf$ are the least essential upper and greatest essential lower bounds, and $∣ \cdot ∣$ denotes Lebesgue measure. The signed-extrema convention is Weak subsolutions and supersolutions of a divergence-form equation; The essential supremum is attained as the least essential bound and The essential supremum of a measurable function with respect to a measure concern the corresponding absolute essential bound.

[F5]

Fatou's lemma: if nonnegative indicators have pointwise lower limit at least the indicator of a limiting set, then the measure of that set is at most the lower limit of the approximating measures (Fatou's lemma).

Proof

technique · direct; after normalisation the measure of the level sets is driven down by a telescoping De Giorgi iteration, and the local boundedness estimate turns the small measure of the top level set into a sup bound

1.1givenF1F2F4algebra

Dichotomy and normalisation. For any $B_{R} (x_{0}) ⋐ Ω$ , local boundedness applied to $u^{+}$ and $(- u)^{+}$ on slightly larger interior balls gives finite $M, m$ ; these truncations are subsolutions by [F2]. The strict sets ${u > (M + m) / 2} \cap B_{R}$ and ${u < (M + m) / 2} \cap B_{R}$ are disjoint, so at least one has measure at most $∣ B_{R} ∣ / 2$ , proving claim 1. For claim 2 assume now $B_{2 R} (x_{0}) ⋐ Ω$ . If $L : = (M - m) / 2 = 0$ , then $u$ is constant a.e. on $B_{R}$ and the reduction is immediate. Otherwise define $v (y) : = (u (x_{0} + R y) - (M + m) / 2) / L$ on $B_{2} (0)$ . It solves the homogeneous equation with rescaled coefficients $A (x_{0} + R y)$ and the same bounds $θ, M_{a}$ , with essential extrema $1, - 1$ on $B_{1}$ . Oscillations scale by $L$ , so it remains to prove ${ess osc}_{B_{1 / 2}} v \leq 2 - η_{0}$ for a universal $η_{0} > 0$ . The dichotomy gives the required half-level measure bound on $B_{1}$ .

1.2givenF3F4F5algebra

The measure estimate. Let $w \in H^{1} (B_{R})$ and $t < T$ . Then $∣ {w \leq t} \cap B_{R} ∣ ∣ {w \geq T} \cap B_{R} ∣^{1 - 1 / n} \leq \frac{C_{n}}{T - t} ∣ B_{R} ∣ \int_{B_{R}} ∣ D w ∣ d x$ . For smooth $w$ , fix $y$ with $w (y) \geq T$ and write $x = y + r ω$ for $x$ with $w (x) \leq t$ . Along the segment, $T - t \leq w (y) - w (x) \leq \int_{0}^{r} ∣ D w (y + s ω) ∣ d s$ . Integrating over the low set in polar coordinates, interchanging the radial integrals, and using $r \leq 2 R$ gives $∣ {w \leq t} \cap B_{R} ∣ \leq \frac{C_{n} ∣ B_{R} ∣}{T - t} \int_{B_{R}} \frac{∣ D w (z) ∣}{∣ z - y ∣^{n - 1}} d z .$ Integrate this in $y$ over $H : = {w \geq T} \cap B_{R}$ . For every measurable $E$ of finite measure, splitting the kernel integral at radius $∣ E ∣^{1 / n}$ gives $\sup_{z} \int_{E} ∣ z - y ∣^{1 - n} d y \leq C_{n} ∣ E ∣^{1 / n}$ ; hence the asserted inequality follows after division by $∣ H ∣^{1 / n}$ (the cases $∣ H ∣ = 0$ or $∣ {w \leq t} ∣ = 0$ are immediate). For general $w$ , choose $w_{j} \in C^{\infty} ({\overline{B}}_{R})$ converging strongly in $H^{1}$ and a subsequence converging a.e. Given $0 < ϵ < (T - t) / 2$ , apply the smooth inequality to $w_{j}$ at levels $t + ϵ, T - ϵ$ . Pointwise lower limits of the indicators dominate those of ${w \leq t}$ and ${w \geq T}$ ; [F5] passes the left side to the limit, while strong $L^{2}$ convergence of gradients gives convergence of $\int ∣ D w_{j} ∣$ . Letting $ϵ ↓ 0$ proves the claim. For its transition-set form, apply it to $z : = \min {(w - t)^{+}, T - t}$ at levels $0, T - t$ . Then ${z \leq 0} = {w \leq t}$ , ${z \geq T - t} = {w \geq T}$ , and $D z = 1_{{t < w < T}} D w$ a.e. Thus if $∣ {w \leq t} \cap B_{R} ∣ \geq γ ∣ B_{R} ∣$ , then Cauchy--Schwarz gives $∣ {w \geq T} \cap B_{R} ∣^{1 - 1 / n} \leq \frac{C_{n}}{γ (T - t)} ∣ {t < w < T} \cap B_{R} ∣^{1 / 2} (\int_{B_{R}} ∣ D (w - t)^{+} ∣^{2})^{1 / 2} .$ The Sobolev truncation chain rule also gives $D w = 0$ a.e. on the endpoint level sets.

2.1step 1.2F2algebra

The telescoping iteration. Work with the normalised $v$ of step 1.1 on $B_{1}$ and suppose first $∣ {v > 0} \cap B_{1} ∣ \leq \frac{1}{2} ∣ B_{1} ∣$ . Set $s : = (3 / 4)^{1 / n}$ , so $∣ B_{s} ∣ = \frac{3}{4} ∣ B_{1} ∣$ , and put $T_{k} : = 1 - 2^{- k - 1}$ , $E_{k} : = {v > T_{k}} \cap B_{s}$ , and $M_{k} : = ∣ E_{k} ∣$ . Since ${v \leq 0} \cap B_{1} = B_{1} ∖ ({v > 0} \cap B_{1})$ has measure at least $\frac{1}{2} ∣ B_{1} ∣$ and $T_{k} > 0$ , it follows that $∣ {v \leq T_{k}} \cap B_{s} ∣ \geq \frac{1}{2} ∣ B_{1} ∣ - ∣ B_{1} ∖ B_{s} ∣ = \frac{1}{4} ∣ B_{1} ∣ = \frac{1}{3} ∣ B_{s} ∣$ for every $k$ . The truncated Caccioppoli estimate of [F2], applied with outer radius $1$ and inner radius $s$ , gives $\int_{B_{s}} ∣ D (v - T_{k})^{+} ∣^{2} \leq C (n, θ, M_{a}) (1 - T_{k})^{2} ∣ B_{1} ∣$ , since $v \leq 1$ a.e. on $B_{1}$ . Apply the transition-set inequality of step 1.2 on $B_{s}$ with $t = T_{k}$ , $T = T_{k + 1}$ , and $γ = 1 / 3$ . As $T_{k + 1} - T_{k} = (1 - T_{k}) / 2$ , the level gap cancels the Caccioppoli factor and yields $M_{k + 1}^{1 - 1 / n} \leq C (n, θ, M_{a}) ∣ B_{1} ∣^{1 / 2} (M_{k} - M_{k + 1})^{1 / 2}, M_{k + 1}^{2 - 2 / n} \leq C (n, θ, M_{a}) (M_{k} - M_{k + 1}) .$ The constant absorbs the fixed volume $∣ B_{1} ∣$ .

3.1step 2.1algebra

Summation. Summing the inequalities of step 2.1 over $k = 0, \dots, N - 1$ and using $M_{k + 1} \geq M_{N}$ gives $N M_{N}^{2 - 2 / n} \leq C \sum_{k = 0}^{N - 1} (M_{k} - M_{k + 1}) = C (M_{0} - M_{N}) \leq C ∣ B_{1} ∣$ , hence $M_{N} \leq C (n, θ, M_{a}) N^{- n / (2 n - 2)} ∣ B_{1} ∣$ for every $N \geq 1$ .

4.1step 3.1F1F2algebra

The top level set is finally small. By [F2], $v_{N} : = (v - T_{N})^{+}$ is a nonnegative subsolution of $L_{0} w = 0$ . Apply the local boundedness estimate [F1] on outer ball $B_{s}$ with inner ratio $(2 s)^{- 1}$ ; since $B_{1 / 2} \subset B_{s}$ and $\int_{B_{s}} v_{N}^{2} \leq M_{N} (1 - T_{N})^{2}$ , this gives ${ess sup}_{B_{1 / 2}} v_{N} \leq C_{1} (n, θ, M_{a}) (∣ B_{s} ∣^{- 1} M_{N})^{1 / 2} (1 - T_{N}) \leq C_{2} N^{- n / (4 n - 4)} (1 - T_{N})$ by step 3.1. Choose $N = N (n, θ, M_{a}) \geq 1$ so large that $C_{2} N^{- n / (4 n - 4)} \leq \frac{1}{2}$ ; then $v \leq T_{N} + \frac{1}{2} (1 - T_{N}) = 1 - η_{0}$ on $B_{1 / 2}$ with $η_{0} : = \frac{1}{2} (1 - T_{N}) > 0$ .

5.1step 1.1step 4.1algebra∎

Conclusion of the reduction. If instead $∣ {v < 0} \cap B_{1} ∣ \leq \frac{1}{2} ∣ B_{1} ∣$ , steps 2.1-4.1 apply verbatim to $- v$ (which is again a solution of the homogeneous equation) and give $v \geq - 1 + η_{0}$ on $B_{1 / 2}$ . In the first case ${ess sup}_{B_{1 / 2}} v \leq 1 - η_{0}$ and ${ess inf}_{B_{1 / 2}} v \geq - 1$ , in the second ${ess sup}_{B_{1 / 2}} v \leq 1$ and ${ess inf}_{B_{1 / 2}} v \geq - 1 + η_{0}$ ; in both cases ${ess osc}_{B_{1 / 2}} v \leq 2 - η_{0}$ . Undoing the affine normalisation of step 1.1 multiplies both oscillations by $L$ and preserves the radius ratio, so ${ess osc}_{B_{R / 2} (x_{0})} u \leq (1 - η_{0} / 2) {ess osc}_{B_{R} (x_{0})} u$ , which is claim 2 with $η : = 1 - η_{0} / 2 \in (0, 1)$ and with $η_{0}$ , hence $η$ , depending only on $n, θ, M_{a}$ .

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Sources

Bozhidar Velichkov, Elliptic PDEs: Teorema di De Giorgi (Universita di Pisa; complete 7-page note, in Italian) (standard reference, not scraped)
Brian Krummel, Consequences of De Giorgi-Nash-Moser (4 March 2016; complete 7-page notes) (standard reference, not scraped)