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A piecewise-smooth coefficient gives an $H^{2}$ solution that is not twice classically differentiable

Example

Assume Countable Choice. On $Ω = (- 1, 1)$ let $a (x) = {\begin{cases} 2 + x, & x \leq 0, \\ 2 + 2 x, & x > 0, \end{cases}$ a continuous, piecewise smooth coefficient with $1 \leq a \leq 4$ and $a \in W^{1, \infty} (- 1, 1)$ whose derivative jumps from $1$ to $2$ at $0$ , and define $u (x) = \int_{0}^{x} \frac{d t}{a (t)}$ , the absolutely continuous primitive with $u (0) = 0$ . Then $u \in H^{2} (- 1, 1)$ , the flux is $a u^{'} \equiv 1$ , and $u$ is a local weak solution of $- (a u^{'})^{'} = 0$ in the sense of Local weak solutions of a divergence-form operator. Explicitly $u^{''} (x) = - \frac{a^{'} (x)}{a (x)^{2}} = {\begin{cases} - \frac{1}{(2 + x)^{2}}, & x < 0, \\ - \frac{2}{(2 + 2 x)^{2}}, & x > 0, \end{cases}$ with one-sided limits $- 1 / 4$ at $0^{-}$ and $- 1 / 2$ at $0^{+}$ ; consequently the one-sided difference quotients of $u^{'}$ at $0$ have the distinct limits $- 1 / 4$ and $- 1 / 2$ , the second classical derivative of $u$ at $0$ does not exist, and $u \notin C^{2} (- 1, 1)$ . Thus the weak $H^{2}$ conclusion of Interior $H^{2}$ regularity for divergence-form equations holds for this Lipschitz coefficient while classical twice differentiability fails: weak $H^{2}$ regularity is genuinely weaker than $C^{2}$ .

Facts & Assumptions

Given: The coefficient $a$ above and the primitive $u (x) = \int_{0}^{x} d t / a (t)$ ; the operator $L u = - (a u^{'})^{'}$ with $a^{11} = a$ , $b = c = 0$ .

[F1]

$u$ is locally absolutely continuous, $u (0) = 0$ , and its a.e. derivative is the integrand: $u^{'} (x) = 1 / a (x)$ for a.e. $x \in (- 1, 1)$ ; more precisely $u (x) = \ln (2 + x) - \ln 2$ for $x \leq 0$ and $u (x) = \frac{1}{2} \ln (1 + x)$ for $x \geq 0$ , and these formulas are differentiable with the stated values of $1 / a$ on each side, agreeing at $0$ with value $1 / 2$ . (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ )

[F2]

A class $u \in H^{1} (Ω)$ is a local weak solution of $L u = f$ on $Ω$ if $a (u, v) = \int_{Ω} f \overline{v}$ for every $v \in C_{c}^{\infty} (Ω)$ ; for $L = - (a u^{'})^{'}$ and $f = 0$ this is $\int_{Ω} a u^{'} \overline{v^{'}} d x = 0$ for every $v \in C_{c}^{\infty} (Ω)$ . (Local weak solutions of a divergence-form operator, Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F3]

The operator $- (a u^{'})^{'}$ is uniformly elliptic with $a^{11} = a$ : $1 \leq a \leq 4$ gives $θ = 1$ and $M_{a} = 4$ , and $b = c = 0$ ; $a$ is continuous and piecewise $C^{1}$ with bounded derivative, hence $a \in W^{1, \infty} (- 1, 1)$ . (Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F4]

For a continuous function $w$ that is $C^{1}$ on each of $(- 1, 0)$ and $(0, 1)$ with bounded one-sided derivatives, integrate $w φ^{'}$ on the two half-intervals. The terms at $0$ cancel because its one-sided values agree; the outer terms vanish for $φ \in C_{c}^{\infty} (- 1, 1)$ . Thus its piecewise derivative is its weak derivative. This argument applies to the coefficient $a$ and to $w = 1 / a$ here. Almost-everywhere differentiability and integrability of the classical derivative alone would not suffice for a general function. (Weak derivative of a locally integrable function)

Verification

1.1F1algebragiven

The explicit primitive. By [F1], for $x < 0$ one has $u (x) = \int_{0}^{x} d t / (2 + t) = \ln (2 + x) - \ln 2$ , and for $x > 0$ one has $u (x) = \int_{0}^{x} d t / (2 + 2 t) = \frac{1}{2} \ln (1 + x)$ ; both formulas give $u (0) = 0$ and both one-sided derivatives at $0$ equal $1 / 2$ , so $u$ is differentiable at $0$ and $u^{'} = 1 / a$ on $(- 1, 1)$ . In particular $u^{'}$ is continuous and $u \in H^{1} (- 1, 1) \cap L^{\infty} (- 1, 1)$ .

1.2F1F4algebra

The second derivative. On $(- 1, 0)$ and on $(0, 1)$ the coefficient is smooth and $u^{''} = - a^{'} / a^{2}$ , namely $- 1 / (2 + x)^{2}$ and $- 2 / (2 + 2 x)^{2}$ respectively; both expressions are bounded in absolute value by $1$ , so $u^{''} \in L^{\infty} (- 1, 1)$ . Moreover $∣ 1 / a (x) - 1 / a (y) ∣ \leq ∣ a (x) - a (y) ∣ \leq 2 ∣ x - y ∣$ because $a \geq 1$ and $a$ is Lipschitz with constant $2$ ; hence $u^{'} = 1 / a$ is Lipschitz and continuous at $0$ . Applying the piecewise test calculation of [F4] shows directly that the displayed bounded piecewise derivative is its weak derivative. Consequently $u \in W^{2, \infty} (- 1, 1) \subseteq H^{2} (- 1, 1)$ , and the displayed formula for $u^{''}$ is the weak second derivative.

2.1F2F3step 1.1algebragiven

The weak equation. Since $a u^{'} \equiv 1$ by step 1.1, for every $v \in C_{c}^{\infty} (- 1, 1)$ one has $\int_{- 1}^{1} a u^{'} \overline{v^{'}} d x = \int_{- 1}^{1} \overline{v^{'}} d x = 0$ , the last integral vanishing because $v$ is compactly supported in $(- 1, 1)$ . Hence the identity of [F2] holds with $f = 0$ ; its hypothesis $u \in H^{1}$ is met by step 1.1. Thus $u$ is the local weak solution of $- (a u^{'})^{'} = 0$ , and by [F3] the operator has $θ = 1$ , $M_{a} = 4$ .

3.1F1step 1.2algebra∎

Failure of classical twice differentiability. By step 1.2 the one-sided limits of $u^{''}$ at $0$ are $- 1 / 4$ at $0^{-}$ and $- 1 / 2$ at $0^{+}$ ; equivalently, the difference quotients of $u^{'}$ have these one-sided limits, since for $h < 0$ one has $(u^{'} (h) - u^{'} (0)) / h = - 1 / (2 (2 + h)) \to - 1 / 4$ and for $h > 0$ one has $(u^{'} (h) - u^{'} (0)) / h = - 1 / (2 + 2 h) \to - 1 / 2$ . Hence $u^{'}$ is not differentiable at $0$ and $u \notin C^{2} (- 1, 1)$ , while $u \in H^{2} (- 1, 1)$ by step 1.2.

Source notes

This is the Lipschitz-coefficient threshold case of the interior $H^{2}$ theorem of Interior $H^{2}$ regularity for divergence-form equations: Teschl's Lemma 10.16 (printed p. 240) assumes exactly $A \in W^{1, \infty}$ , Hunter's Theorem 4.27 (printed p. 112) assumes $C^{1}$ coefficients, and both give $H_{l o c}^{2}$ while saying nothing about $C^{2}$ . The computation above is deliberately self-contained: it verifies $u \in W^{2, \infty} \subseteq H^{2}$ directly from the explicit primitive, so the failure of $C^{2}$ at the derivative jump of $a$ is separated from the regularity theorem rather than resting on it.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter graduate notes) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)

A piecewise-smooth coefficient gives an H2H2 solution that is not twice classically differentiable