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Weak Harnack inequality for nonnegative supersolutions

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 $F \in L_{l o c}^{q} (Ω)$ with $q > n / 2$ . Let $u \in H^{1} (Ω; R)$ satisfy $u \geq 0$ a.e. and $a_{0} (u, φ) \geq - \int_{Ω} F φ d x for every nonnegative φ \in C_{c}^{\infty} (Ω; R),$ i.e. $u$ is a nonnegative weak supersolution of $L_{0} u = - F$ . Then for every ball $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ and every $0 < p < n / (n - 2)$ , $R^{- n / p} ∥ u ∥_{L^{p} (B_{R} (x_{0}))} \leq C ({ess inf}_{B_{R / 2} (x_{0})} u + R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R} (x_{0}))}),$ with $C = C (n, q, θ, M_{a}, p)$ independent of $R$ and $x_{0}$ . For $n = 2$ every finite $p$ is allowed, with the critical Sobolev embedding in place of the $2^{*}$ embedding. The forcing term enters additively and cannot be dropped: the exponent range $0 < p < n / (n - 2)$ and the threshold $q > n / 2$ are the ones the iteration actually produces.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open set $Ω \subseteq R^{n}$ , $n \geq 2$ ; measurable symmetric uniformly elliptic coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with form $a_{0}$ ; a source $F \in L_{l o c}^{q} (Ω)$ , $q > n / 2$ ; a nonnegative $u \in H^{1} (Ω; R)$ with $a_{0} (u, φ) \geq - \int_{Ω} F φ d x$ for all nonnegative $φ \in C_{c}^{\infty} (Ω; R)$ ; a ball $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ and $0 < p < n / (n - 2)$ when $n \geq 3$ , or any finite $p > 0$ when $n = 2$ .

[F1]

Assume the Axiom of Choice. Moser chains for positive supersolutions: if $w \in H^{1} (B_{S}; R)$ satisfies $w > 0$ a.e. and $a_{0} (w, v) \geq 0$ for every nonnegative $v \in H_{0}^{1} (B_{S})$ , then for every $0 < ρ < 1$ and every $p > 0$ , $(\frac{1}{∣ B_{S} ∣} \int_{B_{S}} w^{- p} d x)^{- 1 / p} \leq C_{1} {ess inf}_{B_{ρ S}} w$ , and for some $p_{0} = p_{0} (n, θ, M_{a}) > 0$ one has $(\frac{1}{∣ B_{3 S / 4} ∣} \int_{B_{3 S / 4}} w^{p_{0}} d x)^{1 / p_{0}} \leq C_{1} (\frac{1}{∣ B_{3 S / 4} ∣} \int_{B_{3 S / 4}} w^{- p_{0}} d x)^{- 1 / p_{0}}$ ; the constants depend only on their listed arguments (Moser iteration for positive supersolutions: negative-power and logarithmic comparison, The average of a locally integrable function over a Euclidean ball).

[F2]

Assume the Axiom of Choice. Logarithmic estimate: for every positive supersolution $w$ as in [F1] on a ball, every $η \in C_{c}^{\infty}$ and every $ε > 0$ , $\int η^{2} ∣ D \log (w + ε) ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int ∣ D η ∣^{2} d x$ (Logarithmic Caccioppoli estimate for positive supersolutions).

[F3]

Assume the Axiom of Choice. On the reference ball $B_{2}$ , the weak maximum principle for a zero-trace solution of $L_{0} h = g$ gives $h \geq 0$ when $g \geq 0$ and ${ess sup}_{B_{2}} h \leq C ∥ g^{+} ∥_{L^{q} (B_{2})}$ for $q > n / 2$ (and $q > 1$ in dimension two). The constant is fixed for this ball, and the estimate applies after scaling $B_{2 R}$ to $B_{2}$ (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).

[F4]

Assume the Axiom of Choice. Lax-Milgram and coercivity on $H_{0}^{1} (B_{2})$ : $H^{1} (B_{2})$ is Hilbert by $H^{k}$ is a Hilbert space under the derivative-sum inner product. The closure definition makes $H_{0}^{1} (B_{2})$ a closed linear subspace; a Cauchy sequence converges in $H^{1} (B_{2})$ and its limit remains in that closure, so the inherited inner product makes it Hilbert. The form $a_{0}$ is a bounded coercive form there, and every bounded conjugate-linear functional on $H_{0}^{1} (B_{2})$ is represented by a unique weak Dirichlet solution (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 $H^{- 1} (Ω)$ , Weak Dirichlet solutions for a divergence-form operator, Zero-boundary Sobolev space as a norm closure, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F5]

Assume the Axiom of Choice. Embedding and Holder input: for $n \geq 3$ and $1 \leq r < 2^{*}$ there is $C_{r}$ with $∥ v ∥_{L^{r} (B_{1})} \leq C_{r} ∥ v ∥_{H_{0}^{1} (B_{1})}$ ; in dimension two the same holds for every finite $r$ . By dilation this makes $φ \mapsto \int_{B_{2}} F^{+} φ$ bounded on $H_{0}^{1} (B_{2})$ when $q > n / 2$ (and $q > 1$ for $n = 2$ ), since the conjugate exponent $q^{'}$ lies in the available Sobolev range. Also, if $z \in H^{1} (B_{2})$ , multiplying by a smooth cutoff supported in $B_{2}$ and equal to one on $B_{3 / 2}$ gives $z \in L^{r} (B_{3 / 2})$ for every $1 \leq r < 2^{*}$ when $n \geq 3$ and every finite $r$ when $n = 2$ . The Sobolev norms scale as $∥ v ∥_{L^{r} (B_{R})} \leq C_{r} R^{1 + n / r - n / 2} ∥ D v ∥_{L^{2} (B_{R})}$ for $v \in H_{0}^{1} (B_{R})$ , with the corresponding inhomogeneous local estimate after cutoff; Holder's inequality gives $∥ g ∥_{L^{r} (E)} \leq ∣ E ∣^{1 / r - 1 / s} ∥ g ∥_{L^{s} (E)}$ for $1 \leq r < s \leq \infty$ (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 $L^{p} (μ)$ as the quotient by null functions).

[F6]

Assume the Axiom of Choice. For $U = u + ε \geq ε > 0$ and $0 < s < 1$ , both scalar maps $t \mapsto (\max {t, 0} + ε)^{s - 1}$ and $t \mapsto (\max {t, 0} + ε)^{s / 2}$ are globally Lipschitz. Their compositions with $u$ lie in $H_{l o c}^{1}$ ; the first, multiplied by a compactly supported smooth cutoff squared, gives an $H_{0}^{1}$ test by the product rule, zero extension and smooth density. The second gives $V = U^{s / 2} \in H_{l o c}^{1}$ with $D V = (s / 2) U^{s / 2 - 1} D u$ (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, Compactly supported smooth functions are dense in W^{k,p}(R^n), Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

[F7]

Assume the Axiom of Choice. Dominated convergence passes integrals with an integrable majorant (Dominated convergence). Scaling invariance on doubled balls: with $v (y) : = u (x_{0} + R y)$ and $G (y) : = R^{2} F (x_{0} + R y)$ , the weak supersolution inequality scales to $B_{2}$ , $∥ G ∥_{L^{q} (B_{2})} = R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R} (x_{0}))}$ , and $R^{- n / p} ∥ u ∥_{L^{p} (B_{R} (x_{0}))} = ∣ B_{1} ∣^{1 / p} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} ∣ v ∣^{p})^{1 / p}$ (Uniformly elliptic divergence-form operators and their sesquilinear forms, The average of a locally integrable function over a Euclidean ball, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

Proof

technique · direct; scale the doubled ball to $B_2$, add a Lax-Milgram barrier there to make the solution a homogeneous supersolution on the full region required by Moser's comparison, derive the positive-integrability transitions using only negative-power tests with exponent $s-1<0$, then undo the scaling

1.1givenF3F4F5F7

Removing the source by a barrier on the doubled ball. After the rescaling of [F7] it suffices to treat $R = 1$ , $B_{2} (x_{0}) ⋐ Ω$ , and source norm $∥ F ∥_{L^{q} (B_{2})}$ . Since $F \in L^{q} (B_{2})$ and $q > n / 2$ , the local supersolution inequality extends by density from nonnegative smooth tests to all nonnegative $H_{0}^{1} (B_{2})$ tests: the embedding in [F5] puts $H_{0}^{1} (B_{2})$ in $L^{q^{'}} (B_{2})$ . The functional $φ \mapsto \int_{B_{2}} F^{+} φ d x$ is therefore bounded on $H_{0}^{1} (B_{2})$ , so [F4] gives a unique $h \in H_{0}^{1} (B_{2})$ with $a_{0} (h, φ) = \int_{B_{2}} F^{+} φ d x$ for all $φ \in H_{0}^{1} (B_{2})$ . Since $F^{+} \geq 0$ , [F3] gives $h \geq 0$ ; its radius-two estimate gives ${ess sup}_{B_{2}} h \leq C ∥ F^{+} ∥_{L^{q} (B_{2})} \leq C ∥ F ∥_{L^{q} (B_{2})}$ , where the fixed scaling factor $2^{2 - n / q}$ is absorbed into $C$ . With $w : = u + h$ one has $w \geq 0$ and $a_{0} (w, φ) \geq - \int_{B_{2}} F φ + \int_{B_{2}} F^{+} φ \geq 0$ for every nonnegative $φ \in H_{0}^{1} (B_{2})$ , so $w$ is a nonnegative homogeneous weak supersolution on the full doubled ball. Also $∥ u ∥_{L^{p} (B_{1})} \leq ∥ w ∥_{L^{p} (B_{1})}$ and ${ess inf}_{B_{1 / 2}} w \leq {ess inf}_{B_{1 / 2}} u + {ess sup}_{B_{2}} h$ .

1.2F1F2F5algebra

The seed exponent on a compactly contained ball. Let $w \geq 0$ be a homogeneous weak supersolution on $B_{2}$ and set $U : = w + ε$ . The outer ball $B_{7 / 4}$ is compactly contained in $B_{2}$ , so the comparison clause of [F1], supplied by the logarithmic estimate [F2], gives $(\frac{1}{∣ B_{21 / 16} ∣} \int_{B_{21 / 16}} U^{p_{0}})^{1 / p_{0}} \leq C_{1} (\frac{1}{∣ B_{21 / 16} ∣} \int_{B_{21 / 16}} U^{- p_{0}})^{- 1 / p_{0}}$ for some $p_{0} = p_{0} (n, θ, M_{a}) > 0$ . Apply the negative-power chain of [F1] with outer ball $B_{21 / 16} ⋐ B_{2}$ and ratio $ρ = 8 / 21$ ; it bounds the reciprocal negative moment on $B_{21 / 16}$ by $C_{2} {ess inf}_{B_{1 / 2}} U$ . Decrease the seed to $s_{0} : = \min {p_{0}, 1 / 2} < 1$ ; Jensen's inequality on the normalized ball mean gives $(\frac{1}{∣ B_{21 / 16} ∣} \int_{B_{21 / 16}} U^{s_{0}})^{1 / s_{0}} \leq C_{3} {ess inf}_{B_{1 / 2}} U$ .

1.3givenF3F5F6algebra

The positive-integrability transition for input exponents below one. Fix $0 < s < 1$ and a cutoff $η \in C_{c}^{\infty} (B_{R})$ with $η = 1$ on $B_{r}$ , $0 < r < R \leq 2$ . For $U = w + ε$ , the admissible test $η^{2} U^{s - 1}$ of [F6] and the homogeneous supersolution inequality give $(1 - s) \int η^{2} U^{s - 2} ⟨ A D w, D w ⟩ \leq 2 \int ∣ η ∣ U^{s - 1} ∣ ⟨ A D w, D η ⟩ ∣ .$ Cauchy--Schwarz in the $A$ -energy bounds the right side by $2 M_{a} (\int η^{2} U^{s - 2} ⟨ A D w, D w ⟩)^{1 / 2} (\int U^{s} ∣ D η ∣^{2})^{1 / 2}$ . Absorbing this energy square root and using ellipticity gives $\int η^{2} U^{s - 2} ∣ D w ∣^{2} \leq \frac{4 M_{a}^{2}}{(1 - s)^{2} θ} \int U^{s} ∣ D η ∣^{2}$ . With $V : = U^{s / 2}$ this becomes $\int η^{2} ∣ D V ∣^{2} \leq \frac{M_{a}^{2} s^{2}}{(1 - s)^{2} θ} \int V^{2} ∣ D η ∣^{2}$ . Applying Sobolev to $η V$ and the product rule therefore gives, for $n \geq 3$ and every $1 < λ \leq κ_{*} : = n / (n - 2)$ , or for $n = 2$ and every finite $λ > 1$ , $∥ U ∥_{L^{s λ} (B_{r})} \leq (\frac{C (s, λ, n, θ, M_{a})}{R - r})^{2 / s} ∥ U ∥_{L^{s} (B_{R})} .$ The truncation and density in [F6] justify the test; no estimate for an untruncated positive power is assumed.

2.1step 1.2step 1.3F5algebra

Reaching every exponent in the claimed range. Work on concentric balls between $B_{21 / 16}$ and $B_{1}$ , using equal positive radius gaps for the finitely many transitions below. If $0 < p \leq s_{0}$ , Jensen on $B_{1} \subset B_{21 / 16}$ and step 1.2 give $(\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p} \leq C {ess inf}_{B_{1 / 2}} U$ . For $s_{0} < p \leq κ_{*} s_{0}$ when $n \geq 3$ , use step 1.3 once with $s = s_{0}$ and $λ = p / s_{0} \leq κ_{*}$ . If $κ_{*} s_{0} < p < κ_{*}$ , choose an integer $m \geq 1$ so large that $a : = (p / (κ_{*} s_{0}))^{1 / m} < κ_{*}$ . Apply step 1.3 $m$ times with exponent multiplier $a$ , reaching input exponent $s_{m} = s_{0} a^{m} = p / κ_{*} < 1$ , then once with multiplier $κ_{*}$ . Every input exponent is below one, so all tests in step 1.3 are admissible. The constants are finite and depend only on $n, θ, M_{a}, p$ . In dimension two, for any finite $p > s_{0}$ , a single use of step 1.3 with $s = s_{0}$ and finite $λ = p / s_{0}$ suffices. In every case this proves $(\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p} \leq C {ess inf}_{B_{1 / 2}} U$ for the stated range.

3.1step 2.1F5F7

Removing regularization in the homogeneous case. Let $ε ↓ 0$ in step 2.1. The right side tends to $C {ess inf}_{B_{1 / 2}} w$ , while $U^{p} ↓ w^{p}$ and is dominated by $(w + 1)^{p}$ , integrable on $B_{3 / 2}$ by the cutoff-local Sobolev consequence in [F5] because $p < n / (n - 2) < 2^{*}$ for $n \geq 3$ and $p$ is finite for $n = 2$ (for $p < 1$ , use $(w + 1)^{p} \leq 1 + w$ ). Dominated convergence passes the positive-power mean and yields the homogeneous weak Harnack estimate.

4.1step 1.1step 3.1F5F7algebra∎

Conclusion with the source and the radius rescaling. For the barrier supersolution $w = u + h$ of step 1.1, step 3.1 gives $∥ w ∥_{L^{p} (B_{1})} \leq C {ess inf}_{B_{1 / 2}} w$ . Since $u \leq w$ and ${ess inf}_{B_{1 / 2}} w \leq {ess inf}_{B_{1 / 2}} u + {ess sup}_{B_{2}} h$ , step 1.1 gives $∥ u ∥_{L^{p} (B_{1})} \leq C ({ess inf}_{B_{1 / 2}} u + ∥ F ∥_{L^{q} (B_{2})})$ . Scaling back by [F7] gives the estimate with additive term $R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R})}$ ; the doubled-ball hypothesis supplies the full region used in the barrier and in steps 1.2--2.1. The $n = 2$ argument allows every finite $p$ , and all constants are independent of $R, x_{0}$ .

Remarks

Radius convention. The quantitative interior form of the weak Harnack inequality controls the mean over $B_{R}$ by the essential infimum over $B_{R / 2}$ and requires the supersolution inequality on the doubled ball $B_{2 R}$ , exactly as in Theorem 2 of [K1] and Theorem 2 of [K2]; the statement records this explicitly rather than silently enlarging the class of admissible balls.

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Sources

Brian Krummel, DeGiorgi-Nash lecture notes (15 March 2016; complete 9-page notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete author scan, 118 sheets reproducing the 223 printed pages of the manuscript, two logical pages per sheet) (standard reference, not scraped)
Brian Krummel, Consequences of De Giorgi-Nash-Moser (4 March 2016; complete 7-page notes) (standard reference, not scraped)