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Arbitrary $L^{2}$ boundary data need not have an $H^{1}$ lifting

Statement refuted

Assume the Axiom of Choice. Let $Ω \subset R^{2}$ be a bounded $C^{1}$ domain, let a boundary chart contain the closed straight segment $[0, 1]$ strictly inside its patch, and let $g : = 1_{(0, 1)}$ on that segment, extended by zero. Then $g \in L^{2} (\partial Ω)$ , but $g \notin H^{1 / 2} (\partial Ω) = W^{1 / 2, 2} (\partial Ω)$ : the Slobodeckij seminorm of the line jump at exponent $θ = 1 - 1 / 2$ diverges logarithmically, and by the sharp trace theorem the trace range of $W^{1, 2} (Ω)$ is exactly $W^{1 / 2, 2} (\partial Ω)$ . Consequently no $u \in H^{1} (Ω)$ has $T u = g$ , so the boundary-value problem with this $L^{2}$ datum is not solvable in $H^{1}$ : the lifting hypothesis of the weak Dirichlet formulation cannot be relaxed to arbitrary $L^{2}$ boundary data. No claim is made about the range $1 < p < 2$ , where the same jump function does lie in the trace space.

Facts & Assumptions

Given: The Axiom of Choice together with Countable Choice; a bounded $C^{1}$ domain $Ω \subset R^{2}$ with a boundary chart containing the closed straight segment $[0, 1]$ strictly inside its patch; the jump function $g = 1_{(0, 1)}$ on that segment, extended by zero; the surface measure on $\partial Ω$ ; and the exponent $θ = 1 - 1 / 2 = 1 / 2$ , so that $q : = p θ = 1$ at $p = 2$ . (Bounded C^k domains and boundary charts, Surface integration on compact C1 hypersurfaces, The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The boundary norm of The fractional Sobolev space on a compact $C^{1}$ boundary is a sum over a finite boundary atlas of the Euclidean Slobodeckij norms of the localised representations $(χ_{j} g) \circ Ψ_{j}^{- 1}$ , where $χ_{j}$ is a subordinate finite ambient partition; the Euclidean norm is that of The Gagliardo--Slobodeckij space on Euclidean space, the sum of the $L^{p}$ norm and the extended seminorm $[h]_{s, p} = (\int \int ∣ h (x) - h (y) ∣^{p} ∣ x - y ∣^{- 1 - s p} d x d y)^{1 / p}$ , and the set $W^{s, p} (\partial Ω)$ and its topology are independent of the atlas (Chart independence of the fractional boundary norm).

[F2]

Assume Countable Choice. For nonnegative measurable functions on a product of sigma-finite measure spaces the double integral equals the iterated integrals. (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F3]

A bounded measurable function supported in a set of finite surface measure is an $L^{p} (\partial Ω)$ class for every $p$ . (The space $L^{p} (μ)$ as the quotient by null functions, Surface integration on compact C1 hypersurfaces)

[F4]

Sharp trace theorem: for $1 < p < \infty$ the trace operator $T : W^{1, p} (Ω) \to L^{p} (\partial Ω)$ of The $L^{p}$ trace operator on a bounded $C^{1}$ domain has range exactly $W^{1 - 1 / p, p} (\partial Ω)$ (The sharp trace theorem: boundedness and range in the fractional space, The fractional Sobolev space on a compact $C^{1}$ boundary).

Proof

1.1F3given

The datum is an $L^{2}$ class: the indicator of the straight segment $(0, 1)$ is bounded and is supported in a set of finite surface measure, so by [F3] it is an $L^{p} (\partial Ω)$ class for every $p$ , in particular $g \in L^{2} (\partial Ω)$ . At $p = 2$ the exponent is $θ = 1 / 2$ and $q = p θ = 1$ .

1.2F1F2algebra

The line-jump seminorm: for $h = 1_{(0, 1)}$ on $R$ and $q = p θ > 0$ the integrand $∣ h (x) - h (y) ∣^{p}$ is nonzero exactly when one of $x, y$ lies in $(0, 1)$ and the other does not. By symmetry and [F2] the double integral equals twice its part with $x \in (0, 1)$ , $y \notin (0, 1)$ , and the elementary antiderivative $\int u^{- 1 - q} d u = - u^{- q} / q$ gives $\int_{- \infty}^{0} (x - y)^{- 1 - q} d y + \int_{1}^{\infty} (y - x)^{- 1 - q} d y = \frac{x^{- q} + (1 - x)^{- q}}{q}$ for $x \in (0, 1)$ . Hence $[h]_{θ, p}^{p} = \frac{2}{q} \int_{0}^{1} (x^{- q} + (1 - x)^{- q}) d x = \frac{4}{q} \int_{0}^{1} x^{- q} d x,$ which is finite exactly when $q < 1$ ; translating and scaling the interval $(0, 1)$ to another interval $(a, b)$ changes this value by the finite factor $(b - a)^{1 - q}$ , so finiteness is intrinsic to the interval indicator.

2.1step 1.2algebra

Divergence at $p = 2$ : at $q = 1$ the integral in step 1.2 is $\int_{0}^{1} x^{- 1} d x = + \infty$ , diverging logarithmically at the endpoint $x = 0$ , so $[h]_{1 / 2, 2} = + \infty$ .

3.1F1step 2.1

The boundary norm is infinite: fix the finite atlas of [F1] so that it contains the given straight chart with a cutoff equal to one on the closed segment — possible because the segment lies strictly inside the patch — so that this chart's localised representation is the interval indicator $h$ of step 1.2. Then the corresponding summand of the boundary norm is $+ \infty$ while every other summand is nonnegative, so $∥ g ∥_{W^{1 / 2, 2} (\partial Ω)} = + \infty$ ; by the atlas independence in [F1] the space $W^{1 / 2, 2} (\partial Ω)$ is the same set for every atlas, so $g \notin W^{1 / 2, 2} (\partial Ω) = H^{1 / 2} (\partial Ω)$ .

4.1F4step 3.1∎

No $H^{1}$ lifting: at $p = 2$ the sharp trace theorem identifies the range of $T : W^{1, 2} (Ω) = H^{1} (Ω) \to L^{2} (\partial Ω)$ with $W^{1 / 2, 2} (\partial Ω)$ ; since $g$ lies outside this range, no $u \in H^{1} (Ω)$ satisfies $T u = g$ , and the inhomogeneous problem with this $L^{2}$ datum is not solvable in $H^{1}$ .

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)
Emilio Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284–305 (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer Universitext, 2011, complete 614-page text) (standard reference, not scraped)

Arbitrary L2L2 boundary data need not have an H1H1 lifting