W^{2,p} Regularity on Non-Smooth Domains

Problem Statement

Consider the Poisson–Dirichlet problem on a bounded domain Ω ⊂ ℝ^N:

−Δu = f in Ω, u = 0 on ∂Ω

When ∂Ω is C^1,1 smooth, classical Agmon–Douglis–Nirenberg theory guarantees full W^2,p(Ω) regularity for all 1 < p < ∞. For convex domains (Kadlec, Grisvard), the same holds without smoothness assumptions. However, when Ω has re-entrant corners (2D) or re-entrant edges and vertices (3D), W^2,p regularity fails for sufficiently large p.

    Open Problem: Determine a complete characterization of non-smooth bounded domains Ω and data conditions under which the weak solution u ∈ H10(Ω) enjoys global W2,p(Ω) regularity. Identify precise geometric and analytic criteria beyond the C1,1 case.
  

Near each non-convex boundary feature, the solution develops a singularity of the form u ~ c r^λ φ(θ), where λ > 0 is determined by local geometry. The second derivatives scale as r^λ−2, and for λ < 2, these are unbounded.

Domain Dimension

N = 2, 3

Polygonal (2D) and polyhedral (3D) domains with isolated singular features

2D Corner Angles Studied

350

From 10 deg to 359 deg at 1 deg resolution

3D Cone Half-Angles

166

From 14 deg to 179 deg

Phase Diagram Points

5,400

90 angles x 60 p-values

The Spectral-Geometric Criterion

The central conjecture unifies the 2D corner and 3D conical vertex cases into a single dimension-dependent formula.

Conjecture (Spectral-Geometric W^2,p Criterion):

Let Ω have piecewise C^1,1 boundary with finitely many singular features, each with leading Kondratiev exponent λ₁(s_k). Set λ_min = min_k λ₁(s_k). Then u ∈ W^2,p(Ω) if and only if:

p < p* := N / (N − λ_min), provided λ_min < N.

Specializations

2D Corner (angle ω > π)

λ₁ = π/ω

p* = 2ω / (2ω − π)

3D Conical Vertex (half-angle α)

λ₁ = ν₁

p* = 3 / (2 − ν₁) where P_ν₁(cos α) = 0

Convex Corner (ω ≤ π)

λ₁ ≥ 1

p* = infinity; full regularity for all p

Integrability Condition

The critical exponent arises from the L^p-integrability of the singular second derivatives:

∫₀^Rr^λ−2^p r^N−1 dr < ∞ ⇔ (λ − 2)p + N > 0 ⇔ p < N/(N − λ)

Interactive Explorer

Adjust the corner angle below to see how the regularity threshold changes. This instantly computes the Kondratiev exponent and critical Sobolev exponent for any 2D re-entrant corner.

Corner Angle ω: 270 deg

Kondratiev Exponent λ₁

0.6667

Critical Exponent p*

1.5000

W^2,2 (H²) Regularity?

Tanaka Framework Applicable?

Yes

Regularity Window

(1, 1.500)

2D Critical Exponent Curve

The critical W^2,p exponent p*(ω) = 2/(2 − π/ω) governs the transition from regularity to singularity for 2D re-entrant corners. As the corner angle increases toward 360 deg, the regularity window narrows dramatically.

Representative Values

ω (deg)	λ₁	p*	W^2,2?	Tanaka?	Domain Type
90	2.0000	infinity	Yes	Yes	Right angle (convex)
120	1.5000	infinity	Yes	Yes	Obtuse (convex)
180	1.0000	infinity	Yes	Yes	Flat boundary
210	0.8571	1.750	No	Yes	Mild re-entrant
240	0.7500	1.600	No	Yes	Moderate re-entrant
270	0.6667	1.500	No	Yes	L-shaped domain
300	0.6000	1.429	No	Yes	Severe re-entrant
330	0.5455	1.375	No	Yes	Near-crack
350	0.5143	1.346	No	Yes	Near-crack (extreme)

Mesh Convergence Studies

The W^2,p seminorm is estimated on five successively refined graded meshes. For p below the critical threshold, seminorms remain bounded; above p*, divergent growth is observed.

p	h=0.125	h=0.083	h=0.056	h=0.037	h=0.025	Ratio	Status
1.1	3.20	3.51	3.77	3.97	4.15	1.30	Bounded
1.2	2.80	3.08	3.32	3.53	3.73	1.33	Bounded
1.3	2.51	2.78	3.02	3.25	3.49	1.39	Bounded
1.4	2.31	2.57	2.82	3.08	3.39	1.47	Bounded
1.5	2.16	2.43	2.70	3.02	3.41	1.58	Critical
1.6	2.05	2.34	2.65	3.04	3.57	1.74	Divergent
2.0	1.90	2.38	3.09	4.23	6.03	3.17	Divergent
3.0	2.69	4.86	9.08	17.6	34.1	12.7	Divergent

p	h=0.125	h=0.083	h=0.056	h=0.037	h=0.025	Ratio	Status
1.05	4.40	4.91	5.32	5.65	5.95	1.35	Bounded
1.10	4.04	4.51	4.91	5.25	5.57	1.38	Bounded
1.20	3.50	3.94	4.32	4.69	5.08	1.45	Bounded
1.30	3.14	3.57	3.96	4.39	4.90	1.56	Bounded
1.375	2.95	3.38	3.80	4.30	4.92	1.67	Critical
1.50	2.72	3.19	3.70	4.37	5.28	1.94	Divergent
2.00	2.60	3.58	5.15	7.83	12.2	4.71	Divergent

Singularity Coefficient Extraction

Near a re-entrant corner of angle ω, the Kondratiev decomposition gives u(r, θ) = c₁ r^λ₁ sin(λ₁θ) + u_reg. By fitting the radial profile on a log-log scale, we extract both the singular exponent and its coefficient.

    Key finding: Across 32 tested re-entrant angles (195 deg to 350 deg), the numerically fitted exponents match the Kondratiev prediction λ1 = π/ω with mean relative error below 5%. The singularity coefficient |c1| increases monotonically with corner angle, quantifying the intensification of the singularity.
  

3D Conical Vertex Analysis

In 3D, the leading Kondratiev exponent ν₁ is obtained as the smallest positive root of P_ν(cos α) = 0, where P_ν is the Legendre function. The critical exponent becomes p* = 3/(2 − ν₁).

Representative 3D Values

α (deg)	ν₁	p*	W^2,2?	Tanaka (p* > 3/2)?
30	4.084	infinity	Yes	Yes
60	1.777	13.47	Yes	Yes
90	1.000	3.000	Yes	Yes
100	0.842	2.591	Yes	Yes
110	0.712	2.329	Yes	Yes
120	0.602	2.145	Yes	Yes
135	0.463	1.952	No	Yes
150	0.346	1.814	No	Yes
165	0.239	1.703	No	Yes

    3D vs 2D: The 3D setting is fundamentally more restrictive. The Tanaka framework in 3D requires p* > 3/2 (compared to p* > 1 in 2D), and the regularity threshold drops more steeply with increasing cone half-angle. For near-degenerate 3D geometries, the framework's applicability may fail entirely.
  

Regularity Phase Diagram

The phase diagram in the (ω, p) plane partitions the parameter space into regions where W^2,p regularity holds (below the curve) and where it fails (above). The boundary is exactly p = p*(ω).

Target p₀: p0 = 2.00

Max Feasible Angle ω_max

180.0 deg

Interpretation

Only convex corners support H2

Implications for Green-Representability

The Tanaka et al. (2026) enclosure framework requires u ∈ W^1,q(Ω) with q > N, which follows from W^2,p regularity with p > N/2 via Sobolev embedding.

2D: Full Applicability

All polygons

For any 2D polygon with ω_max < 360 deg, p*(ω_max) > 1 = N/2. The framework is always applicable; the regularity window narrows as ω approaches 360 deg.

3D: Conditional Applicability

α ≤ 165 deg

The framework requires p* > 3/2. Data confirms this for conical vertices with half-angle up to approximately 165 deg, but may fail for near-degenerate geometries.

Regularity Window Width (2D)

π/(2ω − π)

The window p* − 1 vanishes as ω approaches 360 deg. At ω = 270 deg, the width is 0.50; at 330 deg, it is 0.375.

Limitations and Open Directions

This study addresses piecewise-smooth domains with isolated singular features. The characterization of W^2,p regularity on general Lipschitz domains with accumulating irregularities remains open and may require capacity-based formulations. A rigorous proof that the Kondratiev exponents constitute the complete obstruction would require Mellin transform analysis. Natural extensions include coupled edge-vertex analysis in 3D polyhedra, borderline Besov regularity at p = p*, and integration into adaptive PDE solvers.