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Interior regularity does not imply boundary regularity

Statement refuted

Assume Countable Choice. Interior smoothness of a weak solution automatically forces $H^{2}$ up to the boundary of its domain, so that the interior tangential and normal estimates suffice at every boundary point.

Facts & Assumptions

Given: The slit plane $S = C ∖ {x \in R : x \leq 0}$ , the slit disc $Ω = S \cap B_{1} (0) \subset R^{2}$ , the principal square-root biholomorphism $R_{2} (z) = \exp (Log z / 2)$ of the published slit plane theorem, and the function $u (z) = Im R_{2} (z)$ .

[F1]

$R_{2}$ is a biholomorphism from $S$ onto the sector $V_{2} = {r e^{i θ} : r > 0, - π / 2 < θ < π / 2}$ , with inverse $w \mapsto w^{2}$ ; in polar coordinates $z = r e^{i θ}$ with $θ \in (- π, π)$ one has $u (z) = r^{1 / 2} \sin (θ / 2)$ , and $u$ is $C^{\infty}$ and harmonic on $S$ . (A slit-plane root branch biholomorphically parametrizes a sector, The $C^{2}$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair)

[F2]

A class $u \in H^{1} (Ω)$ is a local weak solution of $- Δ u = 0$ on $Ω$ if $\int_{Ω} \nabla u \cdot \overline{\nabla v} d x = 0$ for every $v \in C_{c}^{\infty} (Ω)$ , equivalently for every $v \in H_{0}^{1} (Ω_{2})$ with $Ω_{2} ⋐ Ω$ bounded. (Local weak solutions of a divergence-form operator)

[F3]

Assume Countable Choice. If $a \in W^{1, 2} (U)$ and $b \in W^{1, 2} (U)$ for an open set $U \subseteq R^{n}$ , and at least one of them is compactly supported in $U$ , then $\int_{U} a D_{i} b d x = - \int_{U} b D_{i} a d x$ for every coordinate $i$ , the integrals being bilinear (no conjugation) and absolutely convergent. (Integration by parts for dual-exponent Sobolev functions)

[F4]

For $u = r^{1 / 2} \sin (θ / 2)$ one has $u_{r} = \frac{1}{2} r^{- 1 / 2} \sin (θ / 2)$ and $u_{θ} = \frac{1}{2} r^{1 / 2} \cos (θ / 2)$ , so $∣ \nabla u ∣^{2} = u_{r}^{2} + r^{- 2} u_{θ}^{2} = \frac{1}{4} r^{- 1}$ and $u_{r r} = - \frac{1}{4} r^{- 3 / 2} \sin (θ / 2)$ ; the polar-coordinate formula gives $\int_{Ω} f d x = \int_{0}^{1} \int_{- π}^{π} f (r \cos θ, r \sin θ) r d θ d r$ for every integrable $f$ , the slit lying in the boundary and being ${r > 0, θ = \pm π}$ , a polar-coordinate null set. (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma)

[F5]

The boundary of $Ω$ is not locally the graph of a $C^{1}$ function at $0$ : every neighbourhood of $0$ meets both components of $Ω \cap {x_{2} \neq 0}$ and the slit ${x_{2} = 0, x_{1} < 0}$ lies in $\partial Ω$ , so the defining graph condition of a bounded $C^{k}$ domain fails at $0$ (and along the slit). (Bounded C^k domains and boundary charts)

Counterexample

1.1F1algebragiven

The function is smooth and harmonic inside the domain. By [F1], $R_{2}$ is holomorphic on the slit plane $S \supseteq Ω$ and $u = Im R_{2}$ has polar form $r^{1 / 2} \sin (θ / 2)$ with $θ \in (- π, π)$ ; since a holomorphic function is $C^{\infty}$ in its complex variable, $u \in C^{\infty} (Ω)$ and, by the published component theorem, $u$ is harmonic on $Ω$ : $Δ u = 0$ pointwise.

1.2F4algebra

The function lies in $H^{1}$ . By [F4], $∣ u ∣ \leq r^{1 / 2}$ and $∣ \nabla u ∣^{2} = \frac{1}{4} r^{- 1}$ on $Ω$ , so $\int_{Ω} (∣ u ∣^{2} + ∣ \nabla u ∣^{2}) d x \leq \int_{0}^{1} (r + \frac{1}{4} r^{- 1}) 2 π r d r = 2 π \int_{0}^{1} (r^{2} + \frac{1}{4}) d r$ , which is finite; hence $u \in H^{1} (Ω)$ .

2.1F2F3step 1.1algebra

The function is a local weak solution. Fix $v \in C_{c}^{\infty} (Ω)$ , write $v = v_{1} + i v_{2}$ with real-valued $v_{1}, v_{2}$ , put $d = dist (supp v, \partial Ω) > 0$ and $U = {x : dist (x, supp v) < d / 2}$ , an open bounded set with $\overline{U} \subseteq Ω$ and $supp v \subset U$ . On $U$ the real function $u$ is $C^{\infty}$ by step 1.1, so each real class $\partial_{i} u$ lies in $W^{1, 2} (U)$ , while $v_{1}, v_{2} \in C_{c}^{\infty} (U)$ . Applying [F3] on $U$ with $a = \partial_{i} u$ and $b = v_{j}$ gives $\int_{U} \nabla u \cdot \nabla v_{j} = - \int_{U} v_{j} Δ u = 0$ for each $j = 1, 2$ . Since $\overline{v} = v_{1} - i v_{2}$ , the original pairing is $\int_{U} \nabla u \cdot \nabla v_{1} - i \int_{U} \nabla u \cdot \nabla v_{2} = 0$ . By [F2], $u$ is a local weak solution of $- Δ u = 0$ on $Ω$ .

2.2F4step 1.2algebra

Membership in $H^{2}$ fails at the slit tip. By [F4], For the full Cartesian Hessian $H u = (\partial_{i} \partial_{j} u)_{i, j}$ , $∣ H u ∣ \geq ∣ u_{r r} ∣ = \frac{1}{4} r^{- 3 / 2} ∣ \sin (θ / 2) ∣$ , so $\int_{Ω} ∣ H u ∣^{2} d x \geq \frac{1}{16} \int_{0}^{1} \int_{- π}^{π} r^{- 3} \sin^{2} (θ / 2) r d θ d r = \frac{π}{16} \int_{0}^{1} r^{- 2} d r = + \infty$ ; hence $u \notin H^{2} (Ω)$ and no representative of $u$ is $H^{2}$ up to the boundary point $0$ .

3.1F5step 1.1step 2.2algebra∎

The failure is a boundary phenomenon, not an interior one. Every compactly contained open $Ω^{'} ⋐ Ω$ has positive distance from $0$ and from the slit, and there $u$ is $C^{\infty}$ and harmonic by step 1.1, so $u \in H_{l o c}^{2} (Ω)$ : the interior theorem applies on each $Ω^{'}$ and is not contradicted. The boundary $\partial Ω$ fails the graph condition at $0$ by [F5], so the global boundary hypotheses are unavailable exactly where the $H^{2}$ integral of step 2.2 diverges. Thus interior smoothness does not imply boundary regularity.

Source notes

Teschl's Example 10.1 and the surrounding discussion (printed p. 242) exhibit reentrant boundary points at which the harmonic model function $r^{α} \sin (α θ)$ is in $H^{1}$ but not $H^{2}$ ; the slit disc used here is the limiting case of interior angle $2 π$ with $α = 1 / 2$ , and the square-root biholomorphism of the published slit-plane theorem supplies the harmonicity without any polar-coordinate Laplacian computation. Laugesen's Theorem 5.10 (printed pp. 112-113) states the global boundary estimate under a $C^{2}$ boundary hypothesis, which is exactly what fails here at the slit. The scaffold's earlier witness on the punctured disc was replaced: the function $\log ∣ x ∣$ there is not in $H^{1}$ , so it is not an admissible weak solution and cannot witness the failure; the slit geometry is the minimal correct witness with the same role on this page.

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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)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)