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Critical nonlinearities named in the article

PDF p. 5: paragraph after Corollary 1.6; PDF p. 21: §5.2(B)–(C)

Statement

Assume the Axiom of Choice. The article calls the quintic nonlinearity energy-critical in dimension three at regularity $s = 1$ , and the cubic nonlinearity mass-critical in dimension two at the $L^{2}$ level. These names refer to the scaling of the unforced power Schrödinger equation: mass-critical means its scaling preserves the spatial $L^{2}$ norm, and energy-critical here means that it preserves the spatial $L^{2}$ norm of the gradient, the homogeneous first-order energy-space seminorm. The scaling also dilates the exterior domain; it is not generally a symmetry of one fixed obstacle or of its controlled equation.

For an open spatial set $D \subseteq R^{n}$ and a smooth complex-valued function $f = p + i q$ on $D$ , use Lebesgue measure $d x$ and $∣ f ∣ = (p^{2} + q^{2})^{1 / 2}$ . The spatial norm is $∥ f ∥_{L^{2} (D)} = (\int_{D} ∣ f (x) ∣^{2} d x)^{1 / 2}$ , on almost-everywhere classes when finite. Define the complex gradient componentwise by $\nabla f = \nabla p + i \nabla q = (\partial_{j} p + i \partial_{j} q)_{j = 1}^{n}$ . Its Euclidean length is $∣ \nabla f ∣ = (\sum_{j = 1}^{n} ((\partial_{j} p)^{2} + (\partial_{j} q)^{2}))^{1 / 2}$ , and $∥ \nabla f ∥_{L^{2} (D)} = (\int_{D} ∣ \nabla f (x) ∣^{2} d x)^{1 / 2}$ when finite. The criticality calculation uses these norms for sufficiently smooth solutions with finite displayed norms, at corresponding times on the dilated domains.

Statement references

Linear and nonlinear internally controlled exterior systems

The Axiom of Choice

Complex $L^{p}$ classes and Euclidean test-function conventions

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

Facts & Assumptions

[F1]

The internal system uses $i \partial_{t} u + Δ u + λ ∣ u ∣^{m - 1} u = K h$ , with $m = 3$ for cubic and $m = 5$ for quintic; its exterior domain is $Ω = R^{n} ∖ Θ$ . Linear and nonlinear internally controlled exterior systems

[F2]

The Axiom of Choice gives countable choice: a choice function on the family of sets in a sequence selects an element for each index. The Axiom of Choice The Axiom of Countable Choice (AC_ω)

[F3]

The Euclidean chain rule applies to the real and imaginary components of a smooth complex-valued function and to their derivatives. The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

[F4]

Under countable choice, change of variables for a $C^{1}$ diffeomorphism applies to nonnegative measurable integrands on open sets, with the absolute Jacobian determinant. A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

[F5]

The complex $L^{2}$ convention is $∥ f ∥_{L^{2} (D)} = (\int_{D} ∣ f ∣^{2} d x)^{1 / 2}$ for finite integral, using complex modulus and almost-everywhere classes. Complex $L^{p}$ classes and Euclidean test-function conventions

[F6]

The real scalar gradient is the vector of spatial partial derivatives; applying it to the real and imaginary components defines $\nabla f = \nabla p + i \nabla q$ . The squared length used here is $∣ \nabla f ∣^{2} = \sum_{j = 1}^{n} ((\partial_{j} p)^{2} + (\partial_{j} q)^{2})$ . The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case Complex $L^{p}$ classes and Euclidean test-function conventions

Proof

[F1] The internal system uses $i \partial_{t} u + Δ u + λ ∣ u ∣^{m - 1} u = K h$ , with $m = 3$ for cubic and $m = 5$ for quintic; its exterior domain is $Ω = R^{n} ∖ Θ$ . Linear and nonlinear internally controlled exterior systems

[F2] The Axiom of Choice gives countable choice: a choice function on the family of sets in a sequence selects an element for each index. The Axiom of Choice The Axiom of Countable Choice (AC_ω)

[F3] The Euclidean chain rule applies to the real and imaginary components of a smooth complex-valued function and to their derivatives. The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

[F4] Under countable choice, change of variables for a $C^{1}$ diffeomorphism applies to nonnegative measurable integrands on open sets, with the absolute Jacobian determinant. A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

Given: The equations and dimensions in the statement, with $h = 0$ , and a sufficiently smooth solution $u$ whose displayed spatial norms are finite. Let $ρ > 0$ and set $a = 2 / (m - 1)$ , $u_{ρ} (t, x) = ρ^{a} u (ρ^{2} t, ρ x)$ , and $Ω_{ρ} = ρ^{- 1} Ω$ .

Goal: Identify the two scaling-invariant norms and explain the domain qualification.

1.1F1F3givenalgebra

By the chain rule, both $\partial_{t} u_{ρ}$ and $Δ u_{ρ}$ are $ρ^{a + 2}$ times the corresponding derivative of $u$ evaluated at $(ρ^{2} t, ρ x)$ , while $∣ u_{ρ} ∣^{m - 1} u_{ρ}$ is $ρ^{a m}$ times the original nonlinearity there. Since $a m = a + 2$ , the unforced equation is preserved on the scaled space-time domain. A zero Dirichlet boundary value is carried from $\partial Ω$ to $\partial Ω_{ρ}$ when the boundary values are meaningful.

2.1F2F3F4F5F6step 1.1givenalgebra∎

At corresponding times, apply change of variables $y = ρ x$ on $Ω_{ρ}$ and use the chain rule for the gradient. This gives $∥ u_{ρ} (t) ∥_{L^{2} (Ω_{ρ})} = ρ^{a - n / 2} ∥ u (ρ^{2} t) ∥_{L^{2} (Ω)}$ and $∥ \nabla u_{ρ} (t) ∥_{L^{2} (Ω_{ρ})} = ρ^{a + 1 - n / 2} ∥ \nabla u (ρ^{2} t) ∥_{L^{2} (Ω)}$ . For $m = 3$ , $a = 1$ , so the first exponent is zero when $n = 2$ . For $m = 5$ , $a = 1 / 2$ , so the second exponent is zero when $n = 3$ . These are respectively the mass-critical and energy-critical classifications defined above. Since $Ω_{ρ} = R^{n} ∖ ρ^{- 1} Θ$ , a fixed obstacle and its control cutoff are generally changed by the dilation; no fixed-domain control or well-posedness conclusion follows from this calculation.

Source notes

The article is cited for its two classifications and their context, at the paragraph after Corollary 1.6 and §5.2(B)–(C). The calculation above is a local explanation of the scaling terminology; the article citation is attribution rather than a premise of the calculation. The terms mass and energy name mathematical norms here and do not assert an experimental measurement.

Depends on

Used by · 1

Prior energy-critical null control on Euclidean space

Dependency tree · 2 levels

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2