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Strong maximum principle for weak elliptic solutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 3$ , let $Ω \subseteq R^{n}$ be connected and open, let $A, L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ with $u \geq 0$ a.e. be a weak solution of $L_{0} u = 0$ on $Ω$ . Then either $u = 0$ a.e. on $Ω$ , or $u > 0$ a.e. on $Ω$ ; moreover the Holder representative $u^{*}$ of De Giorgi-Nash interior Holder regularity for divergence-form equations satisfies: if $u^{*}$ vanishes at one point of $Ω$ , then $u^{*} \equiv 0$ on $Ω$ , and otherwise $u^{*} > 0$ on all of $Ω$ . In particular a nonnegative weak solution that is not identically zero is strictly positive after the representative is fixed, and no interior zero is possible.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a connected open set $Ω \subseteq R^{n}$ , $n \geq 3$ ; uniformly elliptic measurable symmetric coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ ; a nonnegative weak solution $u \in H^{1} (Ω; R)$ of $L_{0} u = 0$ ; a Holder representative $u^{*}$ of $u$ on $Ω$ .

[F1]

Assume the Axiom of Choice. Zero-set propagation: if $u \geq 0$ a.e. is a weak solution of $L_{0} u = 0$ on a connected open set and $u^{*}$ is a continuous representative with $u^{*} (x_{0}) = 0$ for some $x_{0} \in Ω$ , then $u^{*} \equiv 0$ on $Ω$ (Zero-set propagation for a nonnegative Holder weak solution, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[F2]

Assume the Axiom of Choice. The representative exists, is continuous, and agrees with $u$ almost everywhere, so $u^{*} \geq 0$ everywhere because $u \geq 0$ a.e. and $u^{*}$ is continuous; conversely if $u^{*} > 0$ on $Ω$ then $u > 0$ a.e. (De Giorgi-Nash interior Holder regularity for divergence-form equations, Local Hölder and scaled C-two-alpha norms on balls, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Assume the Axiom of Choice. Weak solution vocabulary: $L_{0} u = 0$ weakly means $a_{0} (u, v) = 0$ for every $v \in H_{0}^{1} (Ω)$ , and in particular $u$ is both a weak subsolution and a weak supersolution (Weak subsolutions and supersolutions of a divergence-form equation, Uniformly elliptic divergence-form operators and their sesquilinear forms).

Proof

technique · direct dichotomy; either the continuous representative vanishes somewhere, in which case the zero-set propagation lemma makes it identically zero, or it has no zero, in which case continuity and nonnegativity make it strictly positive

1.1givenF1F2

The case of a zero. If there is $x_{0} \in Ω$ with $u^{*} (x_{0}) = 0$ , then [F1] gives $u^{*} \equiv 0$ on the connected set $Ω$ ; since $u = u^{*}$ a.e., $u = 0$ a.e. on $Ω$ .

2.1step 1.1F2F3∎

The case of no zero. If $u^{*}$ vanishes nowhere on $Ω$ , then $u^{*} \neq 0$ everywhere; by [F2] $u^{*} \geq 0$ everywhere, so $u^{*} > 0$ on all of $Ω$ , and hence $u > 0$ a.e. on $Ω$ . Thus either $u = 0$ a.e. or $u > 0$ a.e.; in the first case $u^{*} \equiv 0$ (as the continuous representative of the zero class) and in the second $u^{*} > 0$ everywhere. In particular no point of $Ω$ can be an interior zero of $u^{*}$ unless $u^{*}$ vanishes identically. All arguments use Countable Choice and the Axiom of Choice only through the suppliers named above.

Depends on

Used by

Degenerate ellipticity allows nonconstant solutions with interior zero sets Counterexample

Dependency tree · two levels

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Sources

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