Study of a singular equation set in the half-space

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Preprint submitted on 5 Oct 2011

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Study of a singular equation set in the half-space

Chérif Amrouche, Fabien Dahoumane, Guy Vallet

To cite this version:

Chérif Amrouche, Fabien Dahoumane, Guy Vallet. Study of a singular equation set in the half-space.

2010. �hal-00629120�

❙t✉❞② ♦❢ ❛ s✐♥❣✉❧❛r ❡q✉❛t✐♦♥ s❡t ✐♥ t❤❡

❤❛❧❢✲s♣❛❝❡

❈✳ ❆♠r♦✉❝❤❡✱ ❋✳ ❉❛❤♦✉♠❛♥❡✱ ●✳ ❱❛❧❧❡t

❏✉❧② ✶✻✱ ✷✵✶✵

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❆♣♣❧✐q✉é❡s ❯▼❘✲❈◆❘❙ ◆➦ ✺✶✹✷

■P❘❆ ❇P ✶✶✺✺

✻✹✵✶✸ P❛✉ ❝❡❞❡①

❝❤❡r✐❢✳❛♠r♦✉❝❤❡❅✉♥✐✈✲♣❛✉✳❢r✱ ❢❛❜✐❡♥✳❞❛❤♦✉♠❛♥❡❅✉♥✐✈✲♣❛✉✳❢r✱ ❣✉②✳✈❛❧❧❡t❅✉♥✐✈✲♣❛✉✳❢r

❆❜str❛❝t✳ ❚❤✐s ✇♦r❦ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ ❛ s✐♥❣✉❧❛r ❡q✉❛t✐♦♥ s❡t

✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡✱ ✇✐t❤ ❛ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t t❤❛t ❜❧♦✇s ✉♣ ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ▼♦r❡

♣r❡❝✐s❡❧②✱ ❢♦r ❛ ❞❛t✉♠g :R³+ →R✱ ♦✉r ♣r♦❜❧❡♠ ❝♦♥s✐sts ✐♥ s❡❡❦✐♥❣ u:R³+ →R

❢♦r♠❛❧❧② s♦❧✉t✐♦♥ t♦✿

−❞✐✈

„ 1

x3

∇u

«

=g✐♥R³+, u= 0♦♥Γ =R²× {0}.

❲❡ ❣✐✈❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss r❡s✉❧ts ♦❢ ✇❡❛❦ ❛♥❞ str♦♥❣ s♦❧✉t✐♦♥s ✐♥ s✉✐t❛❜❧❡

✇❡✐❣❤t❡❞ s♣❛❝❡s✱ ✇❤❡r❡ t❤❡ ✇❡✐❣❤t ❞❡♣❡♥❞s ♦♥x3✳

❑❡② ✇♦r❞s✳ ❊❧❧✐♣t✐❝ ❡q✉❛t✐♦♥✱ s✐♥❣✉❧❛r ❡q✉❛t✐♦♥✱ ❤❛❧❢✲s♣❛❝❡✱ ✇❡✐❣❤t❡❞ s♣❛❝❡s✱

✇❡❛❦ s♦❧✉t✐♦♥s✱ str♦♥❣ s♦❧✉t✐♦♥s✱ ❍❛r❞②✬s ✐♥❡q✉❛❧✐t②✳

❆▼❙ s✉❜❥❡❝t ❝❧❛ss✐❢✐❝❛t✐♦♥s✳ ✸✺◗✸✵✱ ✸✺❇✹✵✱ ✼✻❉✵✺✱ ✸✹❈✸✺

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤✐s ✇♦r❦ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ st✉❞② ♦❢ ❛ s✐♥❣✉❧❛r ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥ s❡t ✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡

R³+✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❢♦r ❛ ❞❛t✉♠g : R³+ → R✱ ♦✉r ♣r♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❧♦♦❦✐♥❣ ❢♦r u:R³₊→R❢♦r♠❛❧❧② s♦❧✉t✐♦♥ t♦✿

−❞✐✈

1 x3∇u

=g ✐♥ R³+, u= 0♦♥Γ =R²× {0}. ✭✶✮

❚♦ ✉♥❞❡rst❛♥❞ t❤❡ ♦r✐❣✐♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥✱ ✇❡ t✉r♥ t♦ t❤❡ ✇♦r❦ ♦❢ ❇r❡s❝❤✱ ●✉✐❧❧é♥✲

●♦♥③á❧❡③ ❛♥❞ ▲❡♠♦✐♥❡ ✐♥ ❬✼❪✱ ✇❤♦ st✉❞② ❛♥ ❛♥❛❧♦❣♦✉s ❡q✉❛t✐♦♥ s❡t ✐♥ r❡❣✉❧❛r ❜♦✉♥❞❡❞

❞♦♠❛✐♥ ω ⊂ R²✳ ●✐✈❡♥ h : ω → R ❛♥❞ g : ω → R✱ t❤❡ ❛✉t❤♦rs ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ψ:ω→R❢♦r♠❛❧❧② s♦❧✉t✐♦♥ t♦✿

−❞✐✈1 h∇Ψ

=g ✐♥ω, Ψ = 0♦♥∂ω. ✭✷✮

❇❛s✐❝❛❧❧②✱ h ✐s ❛ ♣♦s✐t✐✈❡ ▲✐s♣✐❝❤t③✲❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣✱ ❜❡❤❛✈✐♥❣ ❛s t❤❡ ❞✐st❛♥❝❡

❢✉♥❝t✐♦♥δ(x^′) =❞✐st(x^′, ∂ω)✱x^′∈ω✱ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢∂ω✳

✶

❚❤❡ ❛✉t❤♦rs ✐♥ ❬✼❪ ❡①♣❧❛✐♥ ❤♦✇ s✉❝❤ s✐♥❣✉❧❛r ❡q✉❛t✐♦♥ ✭✷✮ ♥❛t✉r❛❧❧② ❛♣♣❡❛rs ✐♥ t❤❡

st✉❞② ♦❢ ♠♦❞❡❧s ✐ss✉❡❞ ❢r♦♠ ♦❝❡❛♥♦❣r❛♣❤②✱ ✉♥❞❡r ❛ ❤②❞r♦st❛t✐❝ ♣r❡ss✉r❡ ❛ss✉♠♣t✐♦♥

❛♥❞ s❡t ✐♥ ❞♦♠❛✐♥s ✇✐t❤ ✈❛♥✐s❤✐♥❣ ❞❡♣t❤✳ ■t ✐s t❤❡ ❝❛s❡ ❢♦r ❡①❛♠♣❧❡ ♦❢ t❤❡ ♣❧❛♥❡t❛r②

❣❡♦str♦♣❤✐❝ ❡q✉❛t✐♦♥ ❬✾❪✱ t❤❡ ✈❡rt✐❝❛❧✲❣❡♦str♦♣❤✐❝ ❡q✉❛t✐♦♥s ❬✽❪✱ ❛♥❞ t❤❡ ❤②❞r♦st❛t✐❝

❙t♦❦❡s ♦r ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❬✶✱ ✸✱ ✺✱ ✶✶✱ ✶✺❪✳ ❇❡❢♦r❡ st❛t✐♥❣ t❤❡✐r ♠❛✐♥ r❡s✉❧t✱

✇❡ ❞❡✜♥❡ t❤❡ s♣❛❝❡

H(ω) =n

Ψ∈L²(ω)/ h^−1/2∇ψ∈L²(ω)²,Ψ = 0♦♥∂ωo ,

❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ♥♦r♠kΨkH(ω)=

h^−1/2∇Ψ _L2(ω)²✳

❚❤❡♦r❡♠ ✶✳✶✳ ▲❡tg ❜❡ s✉❝❤ t❤❛tδh^1/2g∈L²(ω)✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥Ψ

♦❢ ✭✷✮ s✉❝❤ t❤❛t Ψ∈H(ω)❛♥❞✿

kΨkH(ω)6C

δh^1/2g _L2(ω).

▼♦r❡♦✈❡r✱ ✐❢ h^1/2g∈L²(ω),t❤❡♥✿

h^1/2∇(1

h∇Ψ)∈L²(ω)⁴,

h^1/2∇(1 h∇Ψ)

_L2(ω)4

6C h^1/2g

L²(ω).

❚♦ ♦✉r ❦♥♦✇❧❡❞❣❡✱ ♥♦ t❤❡♦r❡t✐❝❛❧ st✉❞② ✇❛s ♠❛❞❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❤❛❧❢✲s♣❛❝❡

R²+✱ ❤♦✇❡✈❡r ✉s❡❢✉❧❧ t♦ ♦❜t❛✐♥ ❛ ❣♦♦❞ ❝♦♠♣r❡❤❡♥s✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢ ❛ ❜♦✉♥❞❡❞

❞♦♠❛✐♥✳ ❚❤✐s ♣❛♣❡r ✐s t❤❡r❡❢♦r❡ ❞❡✈♦t❡❞ t♦ t❤❡ st✉❞② ♦❢ s✉❝❤ ❛ ❝❛s❡✱ s❡❡ ✭✶✮✳ ❲❡

♣r❡s❡♥t t❤❡ ❝❛s❡ ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥ ✸ ♦♥❧② ❢♦r t❡❝❤♥✐❝❛❧ r❡❛s♦♥s✱ ❡①♣❧❛✐♥❡❞ ❧❛t❡r✳

❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ♦✉r ✇♦r❦ ❡♥❛❜❧❡s t♦ ❛❝❤✐❡✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t s✐♠✐❧❛r t♦ ❚❤❡♦r❡♠

✶✳✶✱ ✐♥ ❛ ❝♦♥t❡①t ♦❢ ✇❡✐❣❤t❡❞ s♣❛❝❡s✱ ✇❤❡r❡ t❤❡ ✇❡✐❣❤t ❞❡♣❡♥❞s ♦♥ x3✳ ❇❡❢♦r❡❤❛♥❞✱

✇❡ ✐♥tr♦❞✉❝❡ ♣r❡♠❛t✉r❡❧② t❤❡ s♣❛❝❡ W^1,2

−¹2, x3(R³₊)✱ s❡❡ ✭✶✶✮✱ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ♥♦r♠

❣✐✈❡♥ ❜② ✭✶✷✮✳

❚❤❡♦r❡♠ ✶✳✷✳ ▲❡t g ∈ D^′(R³₊) s✉❝❤ t❤❛t x^3/2₃ g ∈ L²(R³₊)✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛

✉♥✐q✉❡ s♦❧✉t✐♦♥ u ∈ W₋^1,1²

2, x3(R³₊) ♦❢ ✭✶✮✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C s✉❝❤ t❤❛t✿

kukW^1,2

−1 2, x3

(R³+)6C x^3/2₃ g

_L2(R³+).

■❢ ✐♥ ❛❞❞✐t✐♦♥✱√x3g∈L²(R³₊),t❤❡♥

√x3∇( 1

x3∇u)∈L²(R³₊)⁹,

√x3∇( 1 x3∇u)

_L2(

R³+)⁹

6Ck√x3gkL²(R³+).

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❝♦♥s✐❞❡r ❛ ❧❛r❣❡r ❝❧❛ss ♦❢ ❞❛t❛ g ✐♥ s✉✐t❛❜❧❡ ✇❡✐❣❤t❡❞ s♣❛❝❡s✱

❛♥❞ ❣✐✈❡ r❡s✉❧ts ♦❢ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ✇❡❛❦ ❛♥❞ str♦♥❣ s♦❧✉t✐♦♥s t♦ ✭✶✮✱ t❤❛t

✐♥❝❧✉❞❡ ❚❤❡♦r❡♠ ✶✳✷✳

◆♦t❡ t❤❛t t❤❡ ✐❞❡❛s ❝♦♥t❛✐♥❡❞ ✐♥ t❤✐s ✇♦r❦✱ ❝❛♥ ❜❡ ❛❞❛♣t❡❞ t♦ t❤❡ R^N+✲❝❛s❡ ✇❤❡r❡

N >3✳ ■♥ t❤❡R²+✲❝❛s❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ✐♥tr♦❞✉❝❡ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❧♦❣❛r✐t❤♠✐❝ ✇❡✐❣❤t✱

✇❤✐❝❤ ❝♦♠♣❧✐❝❛t❡s ❛ ❜✐t ♠♦r❡ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ r❡s✉❧ts✳ ❚❤✐s ✐s ✇❤② ✇❡ ❧✐♠✐t

♦✉r st✉❞② t♦ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥ ✸✳

❆♥ ♦✉t❧✐♥❡ ♦❢ t❤✐s ♣❛♣❡r ✐s ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ ✇❡ s❡t t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦✱

❜② ✐♥tr♦❞✉❝t✐♥❣ t✇♦ ❦✐♥❞s ♦❢ ✇❡✐❣❤t❡❞ s♣❛❝❡s✳ ❋✐rst❧②✱ ✇❡ ❣✐✈❡ ❛♥ ♦✈❡r✈✐❡✇ ♦❢ t❤❡

✇❡✐❣❤t❡❞ s♣❛❝❡s W_α^1,2(R³₊)✱ s❡❡ ✭✹✮✱ ❡♥❝♦✉♥t❡r❡❞ ✐♥ ✇♦r❦s ❝♦♥❝❡r♥✐♥❣ ❢♦r ❡①❛♠♣❧❡

▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥ st✉❞✐❡❞ ✐♥ R³+✳ ❙❡❝♦♥❞❧②✱ ✇❡ ✇✐❧❧ ❜✉✐❧❞ ❛♥♦t❤❡r ❢❛♠✐❧② ♦❢ ✇❡✐❣❤t❡❞

s♣❛❝❡sW_{α, x}^1,2₃(R³₊)✱ s❡❡ ✭✶✶✮✱ ✇❤❡r❡ t❤❡ ✇❡✐❣❤t ❞❡♣❡♥❞s ♦♥x3✱ ❛♥❞ ❡s♣❡❝✐❛❧❧② ❛❞❛♣t❡❞

t♦ t❤❡ st✉❞② ♦❢ ✭✶✮✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ✇♦r❦ ♦❢ ●r✐s✈❛r❞ ❬✶✷❪✱ ✇❡ ✇✐❧❧ ❞❡✜♥❡ ❛ tr❛❝❡ ♦♣❡r❛t♦r✱

✷

❣✐✈❡ ❞❡♥s✐t② r❡s✉❧ts ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ❍❛r❞②✬s ✐♥❡q✉❛❧✐t✐❡s✳ ❲❡ ✇✐❧❧ ✜♥✐s❤ t❤✐s s❡❝t✐♦♥

✇✐t❤ ❛ ❝♦♠♣❛r❛✐s♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢❛♠✐❧✐❡s ♦❢ ✇❡✐❣❤t❡❞ s♣❛❝❡s✱ ♣r♦✈✐♥❣ ❝♦♥t✐♥✉♦✉s

❛♥❞ ❞❡♥s❡ ❡♠❜❡❞❞✐♥❣s ❜❡t✇❡❡♥ t❤❡♠✳

■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ st✉❞② t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s t♦ ✭✶✮✱ ❜②

❝♦♥s✐❞❡r✐♥❣ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❞❛t❛ g✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ♣r♦✈❡ ✐♥ ❚❤❡♦r❡♠ ✸✳✷ t❤❡

❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ ✭✶✮ ✐♥ W_{α, x}^1,2₃(R³₊)✱ ✇❤✐❧❡ ✐♥ ❚❤❡♦r❡♠

✸✳✹✱ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❛ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ ✭✶✮ ✐s ♦❜t❛✐♥❡❞ ✐♥W_α^1,2(R³₊)✳

❚❤❡ ❧❛st s❡❝t✐♦♥ ♦❢ t❤✐s ♣❛♣❡r ✐s ❞❡✈♦t❡❞ t♦ t❤❡ st✉❞② ♦❢ str♦♥❣ s♦❧✉t✐♦♥s t♦ ✭✶✮✱

❝♦♠♣❧❡t✐♥❣ t❤❡ t❤❡♦r❡t✐❝❛❧ st✉❞② ♦❢ Pr♦❜❧❡♠ ✭✶✮✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ❣✐✈❡ t✇♦ s✐t✉❛t✐♦♥s

✇❤❡r❡ t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✸✳✷ ✐s ✐♥ ❢❛❝t ❛ str♦♥❣ s♦❧✉t✐♦♥✱ s❡❡ ❚❤❡♦r❡♠ ✹✳✶

❛♥❞ ❚❤❡♦r❡♠ ✹✳✺✳

✷ ❋✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦

❚❤✐s ✜rst s❡❝t✐♦♥ ✐s ❛✐♠❡❞ t♦ s❡t t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ❛❞♣❛t❡❞ t♦ t❤❡ st✉❞② ♦❢

Pr♦❜❧❡♠ ✭✶✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❞❡✜♥❡ ❛♥❞ st✉❞② ❤❡r❡ t❤❡ ❛♣♣r♦♣r✐❛t❡ s♣❛❝❡ ❢♦r ✇❡❛❦

s♦❧✉t✐♦♥s t♦ ✭✶✮✳

▲❡t ✉s ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❞❛t✉♠ g s✉❝❤ t❤❛tx3g ∈ L²(R³₊)✳ ❋♦r♠❛❧❧②✱ ❧❡t ✉s ♠✉❧t✐♣❧② ✭✶✮ ❜② x3u ❛♥❞ ✐♥t❡❣r❛t❡ ♦✈❡r R³+✳ ❚❤❡♥✱ ✉s✐♥❣

●r❡❡♥✬s ❢♦r♠✉❧❛ ❛♥❞ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ♦✈❡r u✱ ✇❡ ❞❡❞✉❝❡

t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♥❡r❣② ❡q✉❛❧✐t②✿

✂

R³+

|∇u|² dx+1 2

✂

R³+

u x3

2

dx=

✂

R³+

x3gu dx.

❚❤❡r❡❢♦r❡✱ ✇❡ ❛r❡ ♥❛t✉r❛❧❧② ❧❡❛❞ t♦ ❧♦♦❦ ❢♦r ✇❡❛❦ s♦❧✉t✐♦♥sus✉❝❤ t❤❛t✿

∇u∈L²(R³₊)³, u

x3 ∈L²(R³₊), u= 0♦♥Γ. ✭✸✮

❚❤✐s ✜rst s✐t✉❛t✐♦♥ ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✸✳✶✱ ✇❤❡r❡ ✇❡ ❡st❛❜❧✐s❤ t❤❡ ❡①✐st❡♥❝❡

❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s✉❝❤ ❛ ✇❡❛❦ s♦❧✉t✐♦♥✳ ❚❤❡♥✱ s✐♥❝❡ |x3| 6(1 +|x|²)^1/2✱ ❛♥② ✇❡❛❦

s♦❧✉t✐♦♥ s❛t✐s❢②✐♥❣ ✭✸✮ ✐s ✐♥ ❢❛❝t ✐♥ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡W₀^1,2(R³₊)✱ ✇❤❡r❡✿

W₀^1,2(R³₊) = (

u∈ D^′(R³₊)/ u

(1 +|x|²)^1/2 ∈L²(R³₊)❛♥❞∇u∈L²(R³₊)³ )

.

❆♥❞ t❤❡r❡❢♦r❡✱ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥u= 0♦♥Γ✐s ♠❡❛♥✐♥❣❢✉❧❧✳

▲❡t ✉s r❡❝❛❧❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s W_α^1,2(R³₊)✱ ✉s❡❢✉❧❧ ✐♥ t❤❡

s❡q✉❡❧✱ ❛♥❞ st✉❞② ❛ ♥❡✇ ❢❛♠✐❧❧② ♦❢ ✇❡✐❣❤t❡❞ s♣❛❝❡s✱ ✇❤❡r❡ t❤❡ ✇❡✐❣❤t ❞❡♣❡♥❞s ♦♥x3✱ s❡❡ ✭✶✶✮✳

✷✳✶ ❆♥ ♦✈❡r✈✐❡✇ ♦❢ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s W_α^1,2(R³+)

■♥ ❛ ❣❡♥❡r❛❧ ✇❛②✱ t❤❡ s♣❛❝❡s W_α^1,2(R³₊)✱ α ∈ R✱ ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ s✐♥❝❡ t❤❡②

❛r❡ ❛❞❛♣t❡❞ t♦ t❤❡ st✉❞② ♦❢ s♦♠❡ s❡❝♦♥❞ ♦r❞❡r ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s s❡t ✐♥R³+✱ ❛♥❞ ♠♦r❡

♣r❡❝✐s❡❧②✱ t♦ t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥ ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❉✐r✐❝❤❧❡t ♦r ◆❡✉♠❛♥ ❜♦✉♥❞❛r②

❝♦♥❞✐t✐♦♥s✳ ❲❡ r❡❢❡r ❤❡r❡ t♦ t❤❡ ✇♦r❦ ♦❢ ❆♠r♦✉❝❤❡ ❛♥❞ ◆❡ˇc❛s♦✈❛ ❬✷❪✱ ❇♦✉❧♠❡③❛♦✉❞

❬✻❪✱ ♦r t❤❡ ♦♥❡ ♦❢ ❍❛♥♦✉③❡t ❬✶✸❪✳

▲❡t α ∈ R ❛♥❞ s❡t ❢♦r ❛♥② x ∈ R³+✱ ρ(x) = q

1 +|x|²✱ ✇❤❡r❡ |x| ❞❡♥♦t❡s t❤❡

❡✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢x✳ ❚❤❡♥✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡✿

W_α^1,2(R³₊) =

u∈ D^′(R³₊)/ ρ^α−1u∈L²(R³₊)❛♥❞ρ^α∇u∈L²(R³₊)³ , ✭✹✮

✸

✇❤✐❝❤ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡ ❢♦r t❤❡ ♥♦r♠✿

kukWα^1,2(R³+)= ρ^α−1u

2

L²(R³+)+kρ^α∇uk²L²(R³+)³

1/2

. ✭✺✮

❚❤❡ ✇❡✐❣❤t ρ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t t❤❡ s♣❛❝❡ W_α^1,2(R³₊) s❛t✐s✜❡s t✇♦ ❢✉♥❞❛♠❡♥t❛❧

♣r♦♣❡rt✐❡s✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ D(R³₊)✐s ❞❡♥s❡ ✐♥W_α^1,2(R³₊) ✭s❡❡ ❬✶✸❪✮✳ ❖♥ t❤❡ ♦t❤❡r

❤❛♥❞✱ ✐❢ α6= −1/2✱ ❛♥② ❢✉♥❝t✐♦♥s ♦❢ W_α^1,2(R³₊) s❛t✐s✜❡s ❛ P♦✐♥❝❛ré t②♣❡ ✐♥❡q✉❛❧✐t②

✭s❡❡ ❬✷❪✮✳

❆s ¹_ρ ✐s ❜♦✉♥❞❡❞✱ ♥♦t❡ t❤❛t ❢♦r ❛♥② α>β✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥t✐♥✉♦✉s ❡♠❜❡❞❞✐♥❣

❤♦❧❞s✿

W_α^1,2(R³₊)֒→W_β^1,2(R³₊), ✭✻✮

❚❤❡♥✱ ❛♥② ❢✉♥❝t✐♦♥u✐♥W_α^1,2(R³₊)❤❛s ❛ tr❛❝❡ ♦♥Γ✱ ❛♥❞u|^Γ ❜❡❧♦♥❣s t♦ t❤❡ ✇❡✐❣❤t❡❞

s♣❛❝❡Wα^1/2,2(Γ)✳ ❋♦r ♠♦r❡ ♣r❡❝✐s✐♦♥✱ ✇❡ r❡❢❡r t♦ ❬✶✸❪✳

❚❤❡♦r❡♠ ✷✳✶✳ ❚❤❡ ♠❛♣♣✐♥❣γ:u7→u(x^′,0) ❞❡✜♥❡❞ ♦♥D(R³₊)✱ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ✐♥

❛ ✉♥✐q✉❡ ✇❛② t♦ ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣✱ st✐❧❧ ❞❡♥♦t❡❞ ❜②γ✱ ❢r♦♠W_α^1,2(R³₊)

✐♥t♦Wα^1/2,2(Γ)✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ♣❛r❛❣r❛♣❤ ✐s ❞❡❞✐❝❛t❡❞ t♦ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s♣❛❝❡W˚_α^1,2(R³₊)✱

t❤❡ ❝❧♦s✉r❡ ♦❢D(R³₊)✐♥W_α^1,2(R³₊)✳ ❋✐rst❧②✱ ✇❤❡♥α6=−1/2❛♥❞ ❛❝❝♦r❞✐♥❣ ❢♦r ❡①❛♠♣❧❡

t♦ ❬✻❪✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❍❛r❞② t②♣❡ ✐♥❡q✉❛❧✐t②✿

∀u∈W˚_α^1,2(R³₊), ρ^α−1u

L²(R³+)6H^αkρ^α∇ukL²(R³+)³, ✭✼✮

✇❤❡r❡H^α✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✱ ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥α✳ ❆♥❞ ❢♦❧❧♦✇✐♥❣ ❢♦r ❡①❛♠♣❧❡ t❤❡

♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ■✳✷ ♦❢ ❬✶✸❪✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ H^α = _(2α+1)² ✇❤❡♥0 6α61✱ ②✐❡❧❞✐♥❣

✜♥❛❧❧②✿

∀u∈W˚_α^1,2(R³₊), ρ^α−1u

L²(R³+)6 2

(2α+ 1)kρ^α∇ukL²(R³+)³. ✭✽✮

❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭✼✮ ❛♥❞ st✐❧❧ ✐❢ α 6= −1/2✱ t❤❡ ♠❛♣♣✐♥❣ u 7→ kρ^α∇ukL²(R³+)³

❞❡✜♥❡s ❛ ♥♦r♠ ♦♥W˚_α^1,2(R³₊)✱ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♦♥❡ ❞❡✜♥❡❞ ✐♥ ✭✺✮✳ ❙❡❝♦♥❞❧②✱ ✐t ✐s ❛❧s♦

♣r♦✈❡❞ ✐♥ ❬✶✸❪ ♣❛❣❡ ✷✸✽✱ t❤❛t ❢♦r ❛♥②α, β∈Rt❤❡ ♠❛♣♣✐♥❣ ❜❡❧♦✇ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✿

Tρ,β:u∈W˚_α^1,2(R³₊)7→ρ^βu∈W˚_α^1,₋²_β(R³₊), ✭✾✮

❛♥❞ ♦♥❡ ❤❛s t❤❡ ❡st✐♠❛t❡✿

∀u∈W˚_α^1,2(R³₊), kTρ, βukW_α−β^1,2(R³+)6p

1 +β²kukWα^1,2(R³+). ✭✶✵✮

❋✐♥❛❧❧②✱ ❛♥❞ ❛❝❝♦r❞✐♥❣ t♦ ❬✶✸❪ ♣❛❣❡ ✷✺✽ ❚❤❡♦r❡♠ ■■✳ ✸✱ t❤❡ s♣❛❝❡ W˚_α^1,2(R³₊) ❝❛♥ ❜❡

❝❤❛r❛❝t❡r✐③❡❞ ❛s t❤❡ ❦❡r♥❡❧ ♦❢γ✿

W˚_α^1,2(R³₊) =

u∈W_α^1,2(R³₊)/ u= 0♦♥Γ .

❲❡ ✜♥✐s❤ t❤✐s ♣❛r❛❣r❛♣❤ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ s♣❛❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥sW₋⁻_α^1,2(R³₊)❛s t❤❡

❞✉❛❧ s♣❛❝❡ ♦❢W˚_α^1,2(R³₊)✳

✹

✷✳✷ ❚❤❡ s♣❛❝❡s W^1,2

α, x3(R³+)

❋♦rα∈R✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❍✐❧❜❡rt s♣❛❝❡✿

W_{α, x}^1,2₃(R³₊) =

u∈ D^′(R³₊)/ x^α₃⁻¹u∈L²(R³₊), x^α₃∇u∈L²(R³₊)³ , ✭✶✶✮

❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ♥♦r♠✿

kukW_{α, x}^1,2₃(R³+)= x^α₃⁻¹u

2

L²(R³+)+kx^α₃∇uk²L²(R³+)³

1/2

. ✭✶✷✮

■♥ t❤❡ ❡❛r❧② ✶✾✻✵s✱ ❛ s✐♠✐❧❛r s♣❛❝❡ ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❜② ●r✐s✈❛r❞✱ s❡❡ ❬✶✷❪✳ ■♥ ❤✐s

✇♦r❦✱ ●r✐s✈❛r❞ ❝♦♥s✐❞❡rs t❤❡ ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥sus❛t✐s❢②✐♥❣✿

x^α₃u∈L²(R³₊), x^α₃∇u∈L²(R³₊)³.

❍❡ ❡st❛❜❧✐s❤❡s ❝r✉❝✐❛❧ ♣r♦♣❡rt✐❡s✱ s✉❝❤ ❛s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ tr❛❝❡ ♦♣❡r❛t♦r ❛♥❞ ❞❡♥s✐t② r❡s✉❧ts✳ ❋♦❧❧♦✇✐♥❣ ❤✐s ✐❞❡❛s✱ ✇❡ ❜❡❣✐♥ t❤✐s s❡❝t✐♦♥ ❜② ❡st❛❜❧✐s❤✐♥❣ t❤❛t ✇❤❡♥α <1/2✱

❛♥② ❢✉♥❝t✐♦♥su♦❢W_{α, x}^1,2₃(R³₊)♣♦ss❡ss❡s ❛ tr❛❝❡ ✐♥ L²(R²)✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ♣r♦✈❡ t❤❛t

❛♥② ❢✉♥❝t✐♦♥u♦❢W_{α, x}^1,2₃(R³₊)✐s ❝♦♥t✐♥✉♦✉s ✐♥ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥✱ ❡♥s✉r✐♥❣ t❤❡r❡❢♦r❡

t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❡①♣❡❝t❡❞ tr❛❝❡✳

Pr♦♣♦s✐t✐♦♥ ✷✳✷✳ ▲❡tα <1/2✳ ❚❤❡ ♠❛♣♣✐♥❣u7→γu ❞❡✜♥❡❞ ❜②✿

γu= lim

x3→0u(·, x3),

✐s ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢r♦♠ W_{α, x}^1,2₃(R³₊)✐♥t♦L²(R²)✳

Pr♦♦❢✳ ❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ s❡tE=L²(R²)✳ ❚❤❡♥✱ ❧❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡ ✭^✶✮✿

F =n

v∈L²_(α−1)(0, +∞;E)/ v^′ ∈L²_α(0,+∞;E)o ,

❛♥❞ ♥♦t❡ t❤❛tW_{α, x}^1,2₃(R³₊)❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢F✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢

t❤❡ t❤❡♦r❡♠ ♦❢ ❋✉❜✐♥✐✳ ❚❤❡r❡❢♦r❡✱ ✐❢ ✇❡ ♣r♦✈❡ t❤❛t F ✐s ❝♦♥t✐♥✉♦✉s❧② ❡♠❜❡❞❞❡❞ ✐♥

C⁰([0,+∞[;E)✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❞❡✜♥❡u(0)❛s ✇❡❧❧ ❛sγu∈E ❢♦r ❛♥②u∈W_{α, x}^1,2₃(R³₊)✳

❲❡ ❢♦❝✉s ♥♦✇ ♦♥ ❡st❛❜❧✐s❤✐♥❣ t❤✐s ❡♠❜❡❞❞✐♥❣✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❜❡❧♦✇✱

❛✈❛✐❧❛❜❧❡ ❢♦r ❛♥②v∈F ❛♥❞T >0✿

✂ T

0 kv(t)kE dt6

✂ +∞

0 kv(t)k²E t^2(α−1)dt

^1/2 ✂ T 0

t^−2(α−1)dt

!1/2

,

✂ T

0 kv^′(t)kE dt6

✂ +∞

0 kv(t)k²E t^2αdt

^1/2 ✂ T 0

t^−2αdt

!1/2

,

t❤❛t t❤❡ ♠❛♣♣✐♥❣ v 7→ (v, v^′) ✐s ❝♦♥t✐♥✉♦✉s ❢r♦♠ F ✐♥t♦ L¹_❧♦❝([0,+∞[; E)² ✳ ❆s ❛

❝♦♥s❡q✉❡♥❝❡✱ ♦♥❡ ❤❛s ♣r♦✈❡❞ t❤❛t✿

F ֒→C⁰([0,+∞[;E),

❛♥❞ t❤❡ ♣r♦♦❢ ✐s ✜♥✐s❤❡❞✳

❲❡ ❢♦❧❧♦✇ t❤✐s s❡❝t✐♦♥ ❜② ❡st❛❜❧✐s❤✐♥❣ ❛ ❞❡♥s✐t② r❡s✉❧t✱ ❛❞❛♣t❡❞ ❢r♦♠ ❬✶✷❪✳

✶L²_α(0,+∞;E)❞❡♥♦t❡s t❤❡ s❡t ♦❢ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥sv: ]0,+∞[→Es✉❝❤ t❤❛t✿

✂+∞ 0

kvk²_E t^2αdt <∞.

✺

❚❤❡♦r❡♠ ✷✳✸✳ ❋♦r ❛♥② α ∈ R✱ t❤❡ s♣❛❝❡ D(R³₊) ✐s ❞❡♥s❡ ✐♥ W_{α, x}^1,2₃(R³₊)✳ ❆s ❛

❝♦♥s❡q✉❡♥❝❡✱ ✐❢α <1/2t❤❡ ❢✉♥❝t✐♦♥s ♦❢ W_{α, x}^1,2₃(R³₊)❤❛s ♥✉❧❧ tr❛❝❡ ♦♥ R²✳

Pr♦♦❢✳ ■❢ ✇❡ ♣r♦✈❡ t❤❛tD(R³₊)✐s ❞❡♥s❡ ✐♥W_{α, x}^1,2₃(R³₊)✱ t❤❡♥ ✐t ❢♦❧❧♦✇s ❢r♦♠ Pr♦♣♦s✐t✐♦♥

✷✳✷ t❤❛t ✇❤❡♥❡✈❡r t❤❡ tr❛❝❡ ♦❢u✐s ❞❡✜♥❡❞✱ ✐t ✐s ♥❡❝❡ss❛r✐❧② ❡q✉❛❧ t♦ ✵✳

❊①❝❡♣t✐♦♥❛❧❧② ✐♥ t❤✐s ♣r♦♦❢✱ ✇❡ ❞❡♥♦t❡ ❜②E^′(R³₊)t❤❡ s❡t ♦❢ ❞✐str✐❜✉t✐♦♥s ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt ✐♥R³+✱ ❛♥❞E^′(R³₊)t❤❡ s❡t ♦❢ ❞✐str✐❜✉t✐♦♥s ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt ✐♥R³+✳ ❚❤❡

♣r♦♦❢ ✐s t❤❡♥ ❞✐✈✐❞❡❞ ✐♥t♦ t❤r❡❡ st❡♣s✳

❙t❡♣ ♦♥❡✳ ■t ✐s ❝❧❡❛r t❤❛tD(R³₊)✐s ❞❡♥s❡ ✐♥W_{α, x}^1,2₃(R³₊)∩ E^′(R³₊)✱ s✐♥❝❡ ❛♥② ❢✉♥❝t✐♦♥

✐♥ W_{α, x}^1,2₃(R³₊)∩ E^′(R³₊)❜❡❧♦♥❣s t♦H₀¹(R³₊)∩ E^′(R³₊)✳

❙t❡♣ t✇♦✳ ❲❡ ♣r♦✈❡ t❤❛tW_{α, x}^1,2₃(R³₊)∩ E^′(R³₊)✐s ❞❡♥s❡ ✐♥W_{α, x}^1,2₃(R³₊)∩ E^′(R³₊)✳ ▲❡t u ✐♥ W_{α, x}^1,2₃(R³₊)∩ E^′(R³₊) ❛♥❞ ψ ∈ C¹([0,+∞[)✱ s✉❝❤ t❤❛t 0 6 ψ 61✱ ψ(t) = 0✐❢

t61✱ψ(t) = 1 ✐❢t>2✳ ❚❤❡♥✱ ❧❡t ✉s ❞❡✜♥❡ϕε❜②✿

x∈R³+, ϕε(x) =ψ(x3

ε ),

❛♥❞ s❡tuε=uϕε✳ ❖♥❡ ♣r♦✈❡s ✇✐t❤ ♥♦ ❞✐✣❝✉❧t② t❤❛tuε∈W_{α, x}^1,2₃(R³₊)∩E^′(R³₊)✳ ◆❡①t✱

♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥s✿

✂ +∞

0 ku(t)ϕε(t)−u(t)k²L²(R²)t^2(α−1)dt=

✂ 2ε 0

ψ(t ε)−1

ku(t)k²L²(R²)t^2(α−1)dt 6

✂ 2ε

0 ku(t)k²L²(R²)t^2(α−1)dt.

❲❡ ❞❡❞✉❝❡ ❢r♦♠ ▲❡❜❡s❣✉❡✬s t❤❡♦r❡♠ t❤❛t uε→ u✐♥ L²_α−1(0,+∞, L²(R²))✱ ✇❤❡♥ ε

❣♦❡s t♦ ✵✳ ■t ✐s ❛❧s♦ ❝❧❡❛r t❤❛t ϕε∇u → ∇u✐♥ L²_α(0,+∞, L²(R²)³)✳ ❚❤❡r❡❢♦r❡✱ ✐t r❡♠❛✐♥s t♦ ❡st❛❜❧✐s❤ t❤❛tu∇ϕε→0 ✐♥L²_α(0,+∞, L²(R²)³)❛♥❞ t❤✐s ❢♦❧❧♦✇s ❢r♦♠✿

✂ +∞

0 ku(t)∇ϕε(t)k²L²(R²)³t^2αdt=

✂ 2ε ε

u(t)ψ^′(t ε)

2

L²(R²)

t^2α ε² dt

64

✂ 2ε

ε ku(t)k²L²(R²)t^2(α−1)dt.

❙t❡♣ t❤r❡❡✳ ❲❡ ❝❤❡❦ t❤❛t W_{α, x}^1,2₃(R³₊)∩ E^′(R³₊) ✐s ❞❡♥s❡ ✐♥ W_{α, x}^1,2₃(R³₊)✱ ✉s✐♥❣ t❤❡

✉s✉❛❧ ♣r♦❝❡ss ♦❢ tr✉♥❝❛t✐♦♥✳ ■♥❞❡❡❞✱ ❧❡t ϕ∈C¹(R³₊)s✉❝❤ t❤❛t 0 6ϕ61✱ϕ(x) = 1

✐❢|x|61✱ϕ(x) = 0✐❢|x|>2❛♥❞ ❞❡✜♥❡ ϕk ❜②✿

x∈R³+, ϕk(x) =ϕ(x k).

❚❤❡♥ ❢♦r ❛♥②u∈W_{α, x}^1,2₃(R³₊)✱uk=uϕk❜❡❧♦♥❣s t♦W_{α, x}^1,2₃(R³₊)∩E^′(R³₊)✱ ❛♥❞ ❢♦❧❧♦✇✐♥❣

❙t❡♣ t✇♦✱ ✇❡ ❞❡❞✉❝❡ t❤❛tuk ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦u✐♥ W_{α, x}^1,2₃(R³₊)✳

❘❡♠❛r❦ ✷✳✹✳ ❆❝❝♦r❞✐♥❣ ❢♦r ❡①❛♠♣❧❡ t♦ ❬✶✷❪✱ s❡❡ ❚❤❡♦r❡♠ ✶✳✷ ❛♥❞ ❚❤❡♦r❡♠ ✶✳✸✱

♦❜s❡r✈❡ t❤❛t ❢♦r ❛♥②α6= 1/2 t❤❡ ❢♦❧❧♦✇✐♥❣ ❍❛r❞②✬s ✐♥❡q✉❛❧✐t② ❤♦❧❞s✿

∀u∈W_{α, x}^1,2(R³₊), x^α₃⁻¹u

L²(R³+)6 2

|2α−1|kx^α₃∇ukL²(R³+)³. ✭✶✸✮

▲❡t ✉s ♥♦✇ ✐♥tr♦❞✉❝❡ ❛♥❞ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❞✉❛❧ s♣❛❝❡ t♦ W_{α, x}^1,2₃(R³₊)✳

❉❡✜♥✐t✐♦♥ ✷✳✺✳ ▲❡t ✉s ❞❡♥♦t❡ ❜②W₋⁻_{α, x}^1,2₃(R³₊)t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ W_{α, x}^1,2₃(R³₊)✳

✻