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Degenerate ellipticity allows nonconstant solutions with interior zero sets

Statement refuted

Statement refuted. The strong maximum principle of Strong maximum principle for weak elliptic solutions holds for every divergence-form equation with bounded measurable symmetric coefficient matrix that is positive semidefinite, $\sum_{i, j = 1}^{n} a^{i j} (x) ξ_{i} ξ_{j} \geq 0$ for a.e. $x \in Ω$ and all $ξ \in R^{n}$ : degeneracy of the coefficients does not affect the conclusion.

Counterexample. On $Ω = B_{1} (0) \subset R^{3}$ take $A = diag (1, 0, 0)$ (so the matrix is positive semidefinite but degenerate: the uniform ellipticity inequality fails for $ξ = e_{2}$ ) and $u (x) = x_{2}^{+} : = \max {x_{2}, 0}$ . Then $u$ is Lipschitz, nonnegative and nonconstant on $Ω$ , and its zero set contains the lower half-ball ${x_{2} < 0}$ of positive measure. Its weak gradient is $(0, 1_{{x_{2} > 0}}, 0)$ , so $A \nabla u = 0$ a.e. and the weak identity $\int_{B_{1}} A \nabla u \cdot \nabla v d x = 0$ holds for every $v \in H_{0}^{1} (B_{1})$ — the integral identity of Local weak solutions of a divergence-form operator applied to the degenerate matrix. The analogue of Strong maximum principle for weak elliptic solutions fails: uniform ellipticity is a hypothesis of the theorem.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; the unit ball $B_{1} (0) \subset R^{3}$ ; the coefficient matrix $A = diag (1, 0, 0)$ ; and the function $u (x) = x_{2}^{+}$ .

[F1]

The coefficient matrix is bounded, measurable, symmetric and positive semidefinite, but not uniformly elliptic: $ξ^{T} A ξ = ξ_{1}^{2} \geq 0$ for all $ξ$ , while for $ξ = e_{2}$ one has $ξ^{T} A ξ = 0 < θ ∣ ξ ∣^{2}$ for every $θ > 0$ , so the ellipticity hypothesis of Uniformly elliptic divergence-form operators and their sesquilinear forms fails (The notation $H^{k}$ and the reserved zero-boundary symbol).

[F2]

The function $u (x) = x_{2}^{+}$ is Lipschitz and nonnegative on $B_{1} (0) \subset R^{3}$ , nonconstant, and its zero set contains the lower half-ball ${x \in B_{1} (0) : x_{2} < 0}$ , an open interior set of positive measure; moreover $u (x) > 0$ on the upper half-ball (Weak derivative of a locally integrable function, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[F3]

The weak gradient of $u$ is $D u = (0, 1_{{x_{2} > 0}}, 0)$ a.e. on $B_{1} (0) \subset R^{3}$ , so $u \in H^{1} (B_{1} (0))$ . For $A = diag (1, 0, 0)$ one has $A D u = 0$ a.e., hence the degenerate form $a_{0} (u, v) = \int_{B_{1}} A D u \cdot D v d x$ is zero for every $v \in H_{0}^{1} (B_{1})$ (Weak derivative of a locally integrable function, Local weak solutions of a divergence-form operator).

[F4]

The conclusion that fails: for uniformly elliptic coefficients, a nonnegative weak solution of $L_{0} u = 0$ on a connected open set is either zero a.e. or strictly positive a.e., and its continuous representative has no interior zero unless it vanishes identically (Strong maximum principle for weak elliptic solutions, Zero-set propagation for a nonnegative Holder weak solution).

Counterexample

1.1givenF1F2F3

The degenerate matrix and the function. By [F1] the matrix satisfies the stated boundedness and positive-semidefiniteness but violates uniform ellipticity, and by [F2] the function $u (x) = x_{2}^{+}$ is nonnegative, nonconstant and has a half-ball of interior zeros; by [F3] its weak gradient is $(0, 1_{{x_{2} > 0}}, 0)$ and $A D u = 0$ , so $u \in H^{1} (B_{1} (0))$ .

2.1step 1.1F3

The weak identity holds. For every $v \in H_{0}^{1} (B_{1} (0))$ , the matrix-vector product is $A D u = 0$ a.e.; therefore $\int_{B_{1} (0)} A D u \cdot D v d x = 0$ directly. No distributional derivative of a sign function is involved. Hence $u$ is a weak solution of the degenerate equation in the integral sense of Local weak solutions of a divergence-form operator.

3.1step 2.1F4∎

The strong maximum principle fails. The function of step 1.1 is a nonnegative nonconstant weak solution of the degenerate equation whose zero set contains a half-ball of positive measure, so its a.e.-class is neither zero nor strictly positive and its representative has interior zeros; this contradicts the conclusion of [F4] for uniformly elliptic coefficients. The example therefore shows that uniform ellipticity cannot be dropped from Strong maximum principle for weak elliptic solutions and from the De Giorgi-Nash machinery of the page. All verifications use the explicit function and the cited definitions, with no choice principle beyond the declared Axiom of Choice and Countable Choice.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (author manuscript, version 11 February 2025; complete 392-page archived text) (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)