HAL Id: hal-00206215
https://hal.archives-ouvertes.fr/hal-00206215
Preprint submitted on 16 Jan 2008
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Very weak, generalized and strong solutions to the Stokes system in the half-space
Chérif Amrouche, Sárka Necasová, Yves Raudin
To cite this version:
Chérif Amrouche, Sárka Necasová, Yves Raudin. Very weak, generalized and strong solutions to the Stokes system in the half-space. 2008. �hal-00206215�
❱❡r② ✇❡❛❦✱ ❣❡♥❡r❛❧✐③❡❞ ❛♥❞ str♦♥❣ s♦❧✉t✐♦♥s t♦
t❤❡ ❙t♦❦❡s s②st❡♠ ✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡
❈❤ér✐❢ ❆♠r♦✉❝❤❡
a✱ ➆ár❦❛ ◆❡↔❛s♦✈á
b,∗ ✱ ❨✈❡s ❘❛✉❞✐♥
aa▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❆♣♣❧✐q✉é❡s
❈◆❘❙ ❯▼❘ ✺✶✹✷
❯♥✐✈❡rs✐té ❞❡ P❛✉ ❡t ❞❡s P❛②s ❞❡ ❧✬❆❞♦✉r
■P❘❆✱ ❆✈❡♥✉❡ ❞❡ ❧✬❯♥✐✈❡rs✐té ✕ ✻✹✵✵✵ P❛✉ ✕ ❋r❛♥❝❡
b▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡ ♦❢ t❤❡ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s
➎✐t♥á ✷✺ ✕ ✶✶✺ ✻✼ Pr❛❤❛ ✶ ✕ ❈③❡❝❤ ❘❡♣✉❜❧✐❝
❆❜str❛❝t
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ st✉❞② t❤❡ ❙t♦❦❡s s②st❡♠ ✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡ RN
+✱ ✇✐t❤ N ≥ 2✳
❲❡ ❣✐✈❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss r❡s✉❧ts ✐♥ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❆❢t❡r t❤❡
❝❡♥tr❛❧ ❝❛s❡ ♦❢ t❤❡ ❣❡♥❡r❛❧✐③❡❞ s♦❧✉t✐♦♥s✱ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ str♦♥❣ s♦❧✉t✐♦♥s ❛♥❞
s②♠♠❡tr✐❝❛❧② ✐♥ ✈❡r② ✇❡❛❦ s♦❧✉t✐♦♥s ❜② ♠❡❛♥s ♦❢ ❛ ❞✉❛❧✐t② ❛r❣✉♠❡♥t✳
❑❡② ✇♦r❞s✿ ❙t♦❦❡s ♣r♦❜❧❡♠✱ ❍❛❧❢✲s♣❛❝❡✱ ❲❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s
✶✾✾✶ ▼❙❈✿ ✸✺❏✺✵✱ ✸✺❏✺✺✱ ✸✺◗✸✵✱ ✼✻❉✵✼✱ ✼✻◆✶✵
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣❛♣❡r ✐s t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙t♦❦❡s s②st❡♠
(S+)
−∆u+∇π =f ✐♥RN
+, divu =h ✐♥RN+,
u =g ♦♥Γ =RN−1,
∗ ❚❤❡ ✇♦r❦ ♦❢ t❤❡ s❡❝♦♥❞ ❛✉t❤♦r ✇❛s s✉♣♣♦rt❡❞ ❜② t❤❡ ❣r❛♥t ◆✳ ✷✵✶✵✺✵✵✵✺ ❢r♦♠
t❤❡ ●r❛♥t ❆❣❡♥❝② ♦❢ t❤❡ ❈③❡❝❤ ❘❡♣✉❜❧✐❝ ❛♥❞ ❜② ■♥st✐t✉t✐♦♥❛❧ ❘❡s❡❛r❝❤ P❧❛♥ ♥♦✳
❆❱✵❩✶✵✶✾✵✺✵✸✳
❊♠❛✐❧ ❛❞❞r❡ss❡s✿ ❝❤❡r✐❢✳❛♠r♦✉❝❤❡❅✉♥✐✈✲♣❛✉✳❢r ✭❈❤ér✐❢ ❆♠r♦✉❝❤❡✮✱
♠❛t✉s❅♠❛t❤✳❝❛s✳❝③ ✭➆ár❦❛ ◆❡↔❛s♦✈á ✮✱ ②✈❡s✳r❛✉❞✐♥❅❡t✉❞✳✉♥✐✈✲♣❛✉✳❢r ✭❨✈❡s
❘❛✉❞✐♥✮✳
Pr❡♣r✐♥t s✉❜♠✐tt❡❞ t♦ ❊❧s❡✈✐❡r ❙❝✐❡♥❝❡ ❏✉❧② ✶✾✱ ✷✵✵✼
✇✐t❤ ❞❛t❛ ❛♥❞ s♦❧✉t✐♦♥s ✇❤✐❝❤ ❧✐✈❡ ✐♥ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s✱ ❡①♣r❡ss✐♥❣ ❛t t❤❡ s❛♠❡ t✐♠❡ t❤❡✐r r❡❣✉❧❛r✐t② ❛♥❞ t❤❡✐r ❜❡❤❛✈✐♦r ❛t ✐♥✜♥✐t②✳ ❲❡ ✇✐❧❧ ♥❛t✉r❛❧❧②
❜❛s❡ ♦♥ t❤❡ ♣r❡✈✐♦✉s❧② ❡st❛❜❧✐s❤❡❞ r❡s✉❧ts ♦♥ t❤❡ ❤❛r♠♦♥✐❝ ❛♥❞ ❜✐❤❛r♠♦♥✐❝
♦♣❡r❛t♦rs ✭s❡❡ ❬✺❪✱ ❬✻❪✱ ❬✼❪✱ ❬✽❪✮✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❜❛s✐❝ ✇❡✐❣❤ts
❜❡❝❛✉s❡ t❤❡② ❛r❡ t❤❡ ♠♦st ✉s✉❛❧ ❛♥❞ t❤❡② ❛✈♦✐❞ t❤❡ q✉❡st✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ ❢♦r t❤✐s ♦♣❡r❛t♦r ❛♥❞ s②♠♠❡tr✐❝❛❧② t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❞❛t❛✳ ■♥ ❛
❢♦rt❤❝♦♠✐♥❣ ✇♦r❦✱ ✇❡ ✇✐❧❧ ❝♦♠♣❧❡t❡ t❤❡s❡ r❡s✉❧ts ❢♦r t❤❡ ♦t❤❡r t②♣❡s ♦❢ ✇❡✐❣❤ts
✐♥ t❤✐s ❝❧❛ss ♦❢ s♣❛❝❡s✳
❆♠♦♥❣ t❤❡ ✜rst ✇♦r❦s ♦♥ t❤❡ ❙t♦❦❡s ♣r♦❜❧❡♠ ✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡✱ ✇❡ ❝❛♥ ❝✐t❡
❈❛tt❛❜r✐❣❛✳ ■♥ ❬✶✶❪✱ ❤❡ ❛♣♣❡❛❧s t♦ t❤❡ ♣♦t❡♥t✐❛❧ t❤❡♦r② t♦ ❡①♣❧✐❝✐t❧② ❣❡t t❤❡
✈❡❧♦❝✐t② ❛♥❞ ♣r❡ss✉r❡ ✜❡❧❞s✳ ❋♦r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠ ✭f =0❛♥❞h = 0✮✱
❢♦r ✐♥st❛♥❝❡✱ ❤❡ s❤♦✇s t❤❛t ✐❢g ∈Lp(Γ)❛♥❞ t❤❡ s❡♠✐✲♥♦r♠|g|W1−1/p, p
0 (Γ)<∞✱ t❤❡♥ ∇u∈Lp(RN
+)❛♥❞ π ∈Lp(RN
+)✳
❙✐♠✐❧❛r r❡s✉❧ts ❛r❡ ❣✐✈❡♥ ❜② ❋❛r✇✐❣✲❙♦❤r ✭s❡❡ ❬✶✷❪✮ ❛♥❞ ●❛❧❞✐ ✭s❡❡ ❬✶✹❪✮✱ ✇❤♦
❛❧s♦ ❤❛✈❡ ❝❤♦s❡♥ t❤❡ s❡tt✐♥❣ ♦❢ ❤♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❖♥ t❤❡ ♦t❤❡r
❤❛♥❞✱ ▼❛③✬②❛✲P❧❛♠❡♥❡✈s❦✐✟✙✲❙t✉♣②❛❧✐s ✭s❡❡ ❬✶✽❪✮✱ ✇♦r❦ ✇✐t❤✐♥ t❤❡ s✉✐t❛❜❧❡ s❡t✲
t✐♥❣ ♦❢ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s ❛♥❞ ❝♦♥s✐❞❡r ❞✐✛❡r❡♥t s♦rts ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐✲
t✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❡✐r r❡s✉❧ts ❛r❡ ❧✐♠✐t❡❞ t♦ t❤❡ ❞✐♠❡♥s✐♦♥3❛♥❞ t♦ t❤❡ ❍✐❧❜❡r✲
t✐❛♥ ❢r❛♠❡✇♦r❦ ✐♥ ✇❤✐❝❤ t❤❡② ❣✐✈❡ ❣❡♥❡r❛❧✐③❡❞ ❛♥❞ str♦♥❣ s♦❧✉t✐♦♥s✳ ❚❤✐s ✐s
❛❧s♦ t❤❡ ❝❛s❡ ♦❢ ❇♦✉❧♠❡③❛♦✉❞ ✭s❡❡ ❬✶✵❪✮✱ ✇❤♦ ♦♥❧② ❣✐✈❡s str♦♥❣ s♦❧✉t✐♦♥s✳ ❖t❤✲
❡r✇✐s❡✱ ❛❧✇❛②s ✐♥ ❞✐♠❡♥s✐♦♥3✱ ❜② ❋♦✉r✐❡r ❛♥❛❧②s✐s t❡❝❤♥✐q✉❡s✱ ❚❛♥❛❦❛ ❝♦♥s✐❞✲
❡rs t❤❡ ❝❛s❡ ♦❢ ✈❡r② r❡❣✉❧❛r ❞❛t❛✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✈❡❧♦❝✐t✐❡s ✇❤✐❝❤ ❜❡❧♦♥❣ t♦
Wm+3,22 (R3
+)✱ ✇✐t❤ m≥0✭s❡❡ ❬✶✾❪✮✳
▲❡t ✉s ❛❧s♦ q✉♦t❡✱ ❢♦r t❤❡ ❡✈♦❧✉t✐♦♥ ❙t♦❦❡s ♦r ◆❛✈✐❡r✲❙t♦❦❡s ♣r♦❜❧❡♠s✱ ❋✉❥✐❣❛❦✐✲
▼✐②❛❦❛✇❛ ✭s❡❡ ❬✶✸❪✮✱ ✇❤♦ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❜❡❤❛✈✐♦✉r ✐♥t→+∞❀ ❇♦❝❤❡rs✲
▼✐②❛❦❛✇❛ ✭s❡❡ ❬✾❪✮ ❛♥❞ ❑♦③♦♥♦ ✭s❡❡ ❬✶✼❪✮✱ ❢♦r t❤❡ LN✲❉❡❝❛② ♣r♦♣❡rt②❀ ❯❦❛✐
✭s❡❡ ❬✷✵❪✮✱ ❢♦r t❤❡ Lp✲Lq ❡st✐♠❛t❡s ❛♥❞ ●✐❣❛ ✭s❡❡ ❬✶✺❪✮✱ ❢♦r t❤❡ ❡st✐♠❛t❡s ✐♥
❍❛r❞② s♣❛❝❡s✳
❚❤✐s ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❙❡❝t✐♦♥ ✷ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ♥♦t❛t✐♦♥s✱
❢✉♥❝t✐♦♥❛❧ s❡tt✐♥❣ ❛♥❞ r❡❝❛❧❧s ❛❜♦✉t t❤❡ ❙t♦❦❡s s②st❡♠ ✐♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡✳
■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ ❣✐✈❡ s♦♠❡ r❡s✉❧ts ♦♥ ❤♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠s ✇✐t❤ s✐♥❣✉❧❛r
❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛♥❞ ✇❡ ❝♦♠♣❧❡t❡ t❤❡♠ ❜② ❚❤❡♦r❡♠ ✸✳✺ ✇✐t❤ ❛ ❞❡t❛✐❧❡❞
♣r♦♦❢✱ ✇❤✐❝❤ ✐s ❛ ♠♦❞❡❧ ❢♦r ❛♥❛❧♦❣♦✉s r❡s✉❧ts✳ ■♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ st❛rt ♦✉r st✉❞②
♦❢ t❤❡ ❙t♦❦❡s s②st❡♠ ✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡ ❜② t❤❡ ❝❡♥tr❛❧ ❝❛s❡ ♦❢ ❣❡♥❡r❛❧✐③❡❞ s♦✲
❧✉t✐♦♥s ✇❤✐❝❤ ✐s t❤❡ ♣✐✈♦t ♦❢ t❤✐s ✇♦r❦✳ ■♥ ❙❡❝t✐♦♥ ✺✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ str♦♥❣
s♦❧✉t✐♦♥s ❛♥❞ ❣✐✈❡ r❡❣✉❧❛r✐t② r❡s✉❧ts ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❛t❛✳ ■♥ ❙❡❝t✐♦♥ ✻✱ ✇❡
✜♥❞ ✈❡r② ✇❡❛❦ s♦❧✉t✐♦♥s t♦ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠ ✇✐t❤ s✐♥❣✉❧❛r ❜♦✉♥❞❛r②
❝♦♥❞✐t✐♦♥s✳ ❚❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤✐s ♣❛♣❡r ❛r❡ ❚❤❡♦r❡♠ ✹✳✷ ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞
s♦❧✉t✐♦♥s✱ ❚❤❡♦r❡♠s ✺✳✷ ❛♥❞ ✺✳✻ ❢♦r t❤❡ str♦♥❣ s♦❧✉t✐♦♥s✱ ❚❤❡♦r❡♠s ✻✳✼ ❛♥❞
✻✳✾ ❢♦r t❤❡ ✈❡r② ✇❡❛❦ s♦❧✉t✐♦♥s✳
✷
✷ ◆♦t❛t✐♦♥s✱ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ❛♥❞ ❦♥♦✇♥ r❡s✉❧ts
✷✳✶ ◆♦t❛t✐♦♥s
❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r p > 1✱ ✇❡ ❛❧✇❛②s t❛❦❡ p′ t♦ ❜❡ t❤❡ ❍ö❧❞❡r ❝♦♥❥✉❣❛t❡ ♦❢
p✱ ✐✳❡✳
1 p+ 1
p′ = 1.
▲❡t Ω ❜❡ ❛♥ ♦♣❡♥ s❡t ♦❢ RN✱ N ≥ 2✳ ❲r✐t✐♥❣ ❛ t②♣✐❝❛❧ ♣♦✐♥t x ∈ RN ❛s x= (x′, xN)✱ ✇❤❡r❡x′ = (x1, . . . , xN−1)∈RN−1 ❛♥❞xN ∈R✱ ✇❡ ✇✐❧❧ ❡s♣❡❝✐❛❧❧②
❧♦♦❦ ♦♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲s♣❛❝❡ RN
+ = {x ∈ RN; xN > 0}✳ ❲❡ ❧❡t RN+ ❞❡♥♦t❡
t❤❡ ❝❧♦s✉r❡ ♦❢ RN
+ ✐♥ RN ❛♥❞ ❧❡t Γ = {x ∈ RN; xN = 0} ≡ RN−1 ❞❡♥♦t❡ ✐ts
❜♦✉♥❞❛r②✳ ▲❡t |x| = (x21 +· · ·+x2N)1/2 ❞❡♥♦t❡ t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ x✱ ✇❡
✇✐❧❧ ✉s❡ t✇♦ ❜❛s✐❝ ✇❡✐❣❤ts
̺= (1 +|x|2)1/2 ❛♥❞ lg̺= ln(2 +|x|2).
❲❡ ❞❡♥♦t❡ ❜② ∂i t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ∂
∂xi✱ s✐♠✐❧❛r❧② ∂i2 = ∂i ◦ ∂i = ∂2
∂x2i✱
∂ij2 = ∂i ◦∂j = ∂2
∂xi∂xj
, . . . ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐❢ λ = (λ1, . . . , λN) ∈ NN ✐s ❛
♠✉❧t✐✲✐♥❞❡①✱ t❤❡♥
∂λ =∂1λ1· · ·∂NλN = ∂|λ|
∂xλ11· · ·∂xλNN ✱ ✇❤❡r❡ |λ|=λ1+· · ·+λN.
■♥ t❤❡ s❡q✉❡❧✱ ❢♦r ❛♥② ✐♥t❡❣❡rq✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡s✿
✖Pq ✐s t❤❡ s♣❛❝❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ s♠❛❧❧❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ q❀
✖Pq∆ ✐s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❤❛r♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧s ♦❢ Pq❀
✖Pq∆2 ✐s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❜✐❤❛r♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧s ♦❢Pq❀
✖ A∆q ✐s t❤❡ s✉❜s♣❛❝❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ Pq∆✱ ♦❞❞ ✇✐t❤ r❡s♣❡❝t t♦ xN✱ ♦r
❡q✉✐✈❛❧❡♥t❧②✱ ✇❤✐❝❤ s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥ϕ(x′,0) = 0❀
✖ Nq∆ ✐s t❤❡ s✉❜s♣❛❝❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ Pq∆✱ ❡✈❡♥ ✇✐t❤ r❡s♣❡❝t t♦ xN✱ ♦r
❡q✉✐✈❛❧❡♥t❧②✱ ✇❤✐❝❤ s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥∂Nϕ(x′,0) = 0❀
✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t t❤❡s❡ s♣❛❝❡s ❛r❡ r❡❞✉❝❡❞ t♦ {0}✐❢ q <0✳
❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r s✱ ✇❡ ❞❡♥♦t❡ ❜②[s] t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ s✳
●✐✈❡♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ B✱ ✇✐t❤ ❞✉❛❧ s♣❛❝❡ B′ ❛♥❞ ❛ ❝❧♦s❡❞ s✉❜s♣❛❝❡ X ♦❢ B✱
✇❡ ❞❡♥♦t❡ ❜② B′ ⊥X t❤❡ s✉❜s♣❛❝❡ ♦❢B′ ♦rt❤♦❣♦♥❛❧ t♦ X✱ ✐✳❡✳
B′ ⊥X ={f ∈B′; ∀v ∈X, hf, vi= 0}= (B/X)′.
▲❛st❧②✱ ✐❢ k ∈ Z✱ ✇❡ ✇✐❧❧ ❝♦♥st❛♥t❧② ✉s❡ t❤❡ ♥♦t❛t✐♦♥ {1, . . . , k} ❢♦r t❤❡ s❡t ♦❢
t❤❡ ✜rst k ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✱ ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t t❤✐s s❡t ✐s ❡♠♣t② ✐❢ k ✐s
♥♦♥♣♦s✐t✐✈❡✳
✸
✷✳✷ ❲❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s
❋♦r ❛♥② ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡r m✱ r❡❛❧ ♥✉♠❜❡rs p > 1✱ α ❛♥❞ β✱ ✇❡ ❞❡✜♥❡ t❤❡
❢♦❧❧♦✇✐♥❣ s♣❛❝❡✿
Wα, βm, p(Ω) =
u∈ D′(Ω); 0≤ |λ| ≤k, ̺α−m+|λ|(lg̺)β−1∂λu∈Lp(Ω);
k+ 1 ≤ |λ| ≤m, ̺α−m+|λ|(lg̺)β∂λu∈Lp(Ω)
,
✭✷✳✶✮
✇❤❡r❡
k =
−1 ✐❢ Np +α /∈ {1, . . . , m}, m−N
p −α ✐❢ Np +α ∈ {1, . . . , m}.
◆♦t❡ t❤❛tWα, βm, p(Ω)✐s ❛ r❡✢❡①✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ ✐ts ♥❛t✉r❛❧ ♥♦r♠✿
kukWm, p
α, β(Ω) =
X
0≤|λ|≤k
k̺α−m+|λ|(lg̺)β−1∂λukpLp(Ω)
+ X
k+1≤|λ|≤m
k̺α−m+|λ|(lg̺)β∂λukpLp(Ω)
1/p
.
❲❡ ❛❧s♦ ❞❡✜♥❡ t❤❡ s❡♠✐✲♥♦r♠✿
|u|Wm, p
α, β(Ω) =
X
|λ|=m
k̺α(lg̺)β∂λukpLp(Ω)
1/p
.
❚❤❡ ✇❡✐❣❤ts ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✷✳✶✮ ❛r❡ ❝❤♦s❡♥ s♦ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♣❛❝❡
s❛t✐s✜❡s t✇♦ ❢✉♥❞❛♠❡♥t❛❧ ♣r♦♣❡rt✐❡s✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ DRN+ ✐s ❞❡♥s❡ ✐♥
Wα, βm, p(RN+)✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ❢♦❧❧♦✇✐♥❣ P♦✐♥❝❛ré✲t②♣❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s
✐♥Wα, βm, p(RN
+) ✭s❡❡ ❬✺❪✱ ❚❤❡♦r❡♠ ✶✳✶✮✿ ✐❢
N
p +α /∈ {1, . . . , m} ♦r (β−1)p6=−1, ✭✷✳✷✮
t❤❡♥ t❤❡ s❡♠✐✲♥♦r♠ | · |Wm, p
α, β(RN
+) ❞❡✜♥❡s ♦♥ Wα, βm, p(RN+)/Pq∗ ❛ ♥♦r♠ ✇❤✐❝❤ ✐s
❡q✉✐✈❛❧❡♥t t♦ t❤❡ q✉♦t✐❡♥t ♥♦r♠✱
∀u∈Wα, βm, p(RN+), kukWm, p
α, β(RN
+)/Pq∗ ≤C|u|Wm, p
α, β(RN
+), ✭✷✳✸✮
✇✐t❤ q∗ = inf(q, m −1)✱ ✇❤❡r❡ q ✐s t❤❡ ❤✐❣❤❡st ❞❡❣r❡❡ ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧s
❝♦♥t❛✐♥❡❞ ✐♥ Wα, βm, p(RN+)✳ ◆♦✇✱ ✇❡ ❞❡✜♥❡ t❤❡ s♣❛❝❡
W◦m, pα, β(RN
+) = D(RN+)k·kWα, βm, p(RN+);
✇❤✐❝❤ ✇✐❧❧ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ✐♥ ▲❡♠♠❛ ✷✳✷ ❛s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ♥✉❧❧
tr❛❝❡s ✐♥Wα, βm, p(RN
+)✳ ❋r♦♠ t❤❛t✱ ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡W−α,−m, p−β′(RN
+)❛s t❤❡
✹
❞✉❛❧ s♣❛❝❡ ♦❢W◦m, pα, β(RN
+)✳ ■♥ ❛❞❞✐t✐♦♥✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ✭✷✳✷✮✱| · |Wm, p
α, β(RN
+)
✐s ❛ ♥♦r♠ ♦♥W◦m, pα, β(RN
+)✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❢✉❧❧ ♥♦r♠k · kWm, p
α, β(RN
+)✳ ❲❡
✇✐❧❧ ♥♦✇ r❡❝❛❧❧ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡sWα, βm, p(RN
+)✳ ❲❡
❤❛✈❡ t❤❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ✐♠❜❡❞❞✐♥❣s✿
Wα, βm, p(RN
+)֒→Wα−1, βm−1, p(RN
+)֒→ · · ·֒→Wα−m, β0, p (RN
+) ✐❢ N
p +α /∈ {1, . . . , m}.
❲❤❡♥ N
p +α=j ∈ {1, . . . , m}✱ t❤❡♥ ✇❡ ❤❛✈❡✿
Wα, βm, p ֒→ · · ·֒→Wα−j+1, βm−j+1, p֒→Wα−j, β−1m−j, p ֒→ · · ·֒→Wα−m, β−10, p .
◆♦t❡ t❤❛t ✐♥ t❤❡ ✜rst ❝❛s❡✱ ❢♦r ❛♥② γ ∈ R s✉❝❤ t❤❛t N
p +α−γ /∈ {1, . . . , m}
❛♥❞ m∈N✱ t❤❡ ♠❛♣♣✐♥❣
u∈Wα, βm, p(RN+)7−→̺γu∈Wα−γ, βm, p (RN+)
✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ■♥ ❜♦t❤ ❝❛s❡s ❛♥❞ ❢♦r ❛♥② ♠✉❧t✐✲✐♥❞❡①λ∈NN✱ t❤❡ ♠❛♣♣✐♥❣
u∈Wα, βm, p(RN
+)7−→∂λu∈Wα, βm−|λ|, p(RN
+)
✐s ❝♦♥t✐♥✉♦✉s✳ ❋✐♥❛❧❧②✱ ✐t ❝❛♥ ❜❡ r❡❛❞✐❧② ❝❤❡❝❦❡❞ t❤❛t t❤❡ ❤✐❣❤❡st ❞❡❣r❡❡ q ♦❢
t❤❡ ♣♦❧②♥♦♠✐❛❧s ❝♦♥t❛✐♥❡❞ ✐♥ Wα, βm, p(RN
+)✐s ❣✐✈❡♥ ❜②
q=
m− N p +α
!
−1, ✐❢
N
p +α ∈ {1, . . . , m} ❛♥❞ (β−1)p≥ −1,
♦r
N
p +α∈ {j ∈Z; j ≤0} ❛♥❞ βp≥ −1,
"
m− N p +α
!#
, ♦t❤❡r✇✐s❡.
✭✷✳✹✮
❘❡♠❛r❦ ✷✳✶✳ ■♥ t❤❡ ❝❛s❡ β = 0✱ ✇❡ s✐♠♣❧② ❞❡♥♦t❡ t❤❡ s♣❛❝❡ Wα,m, p0 (Ω) ❜② Wαm, p(Ω)✳ ■♥ ❬✶✻❪✱ ❍❛♥♦✉③❡t ✐♥tr♦❞✉❝❡❞ ❛ ❝❧❛ss ♦❢ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s
✇✐t❤♦✉t ❧♦❣❛r✐t❤♠✐❝ ❢❛❝t♦rs✱ ✇✐t❤ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥✳ ❲❡ r❡❝❛❧❧ ❤✐s ❞❡✜♥✐t✐♦♥
✉♥❞❡r t❤❡ ♥♦t❛t✐♦♥ Hαm, p(Ω)✿
Hαm, p(Ω) =
u∈ D′(Ω); 0≤ |λ| ≤m, ̺α−m+|λ|∂λu∈Lp(Ω)
.
■t ✐s ❝❧❡❛r t❤❛t ✐❢ N
p +α /∈ {1, . . . , m}✱ ✇❡ ❤❛✈❡ Wαm, p(Ω) = Hαm, p(Ω)✳ ❚❤❡
❢✉♥❞❛♠❡♥t❛❧ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢❛♠✐❧✐❡s ♦❢ s♣❛❝❡s ✐s t❤❛t t❤❡ ❛s✲
s✉♠♣t✐♦♥ ✭✷✳✷✮ ❛♥❞ t❤✉s t❤❡ P♦✐♥❝❛ré✲t②♣❡ ✐♥❡q✉❛❧✐t② ✭✷✳✸✮✱ ❤♦❧❞ ❢♦r ❛♥② ✈❛❧✉❡
♦❢ (N, p, α) ✐♥ Wαm, p(Ω)✱ ❜✉t ♥♦t ✐♥ Hαm, p(Ω) ✐❢ N
p +α∈ {1, . . . , m}✳
✺
✷✳✸ ❚❤❡ s♣❛❝❡s ♦❢ tr❛❝❡s
■♥ ♦r❞❡r t♦ ❞❡✜♥❡ t❤❡ tr❛❝❡s ♦❢ ❢✉♥❝t✐♦♥s ♦❢Wαm, p(RN
+)✭❤❡r❡ ✇❡ ❞♦♥✬t ❝♦♥s✐❞❡r t❤❡ ❝❛s❡β 6= 0✮✱ ❢♦r ❛♥②σ ∈ ]0,1[✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡✿
W0σ, p(RN) =
(
u∈ D′(RN); w−σu∈Lp(RN) ❛♥❞
Z
RN×RN
|u(x)−u(y)|p
|x−y|N+σp ❞x❞y <∞
)
,
✭✷✳✺✮
✇❤❡r❡ w = ̺ ✐❢ N/p 6= σ ❛♥❞ w = ̺(lg̺)1/σ ✐❢ N/p = σ✳ ■t ✐s ❛ r❡✢❡①✐✈❡
❇❛♥❛❝❤ s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ ✐ts ♥❛t✉r❛❧ ♥♦r♠✿
kukWσ, p
0 (RN) =
u wσ
p Lp(RN)
+
Z
RN×RN
|u(x)−u(y)|p
|x−y|N+σp ❞x❞y
!1/p
.
❙✐♠✐❧❛r❧②✱ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡r α∈R✱ ✇❡ ❞❡✜♥❡ t❤❡ s♣❛❝❡✿
Wασ, p(RN) =
(
u∈ D′(RN); wα−σu∈Lp(RN),
Z
RN×RN
|̺α(x)u(x)−̺α(y)u(y)|p
|x−y|N+σp ❞x❞y <∞
)
,
✇❤❡r❡ w=̺ ✐❢ N/p+α 6=σ ❛♥❞ w=̺(lg̺)1/(σ−α) ✐❢ N/p+α =σ✳ ❋♦r ❛♥② s∈R+✱ ✇❡ s❡t
Wαs, p(RN) =
u∈ D′(RN); 0≤ |λ| ≤k, ̺α−s+|λ|(lg̺)−1∂λu∈Lp(RN);
k+ 1≤ |λ| ≤[s]−1, ̺α−s+|λ|∂λu∈Lp(RN); |λ|= [s], ∂λu∈Wασ, p(RN)
,
✇❤❡r❡ k = s−N/p−α ✐❢ N/p+α ∈ {σ, . . . , σ+ [s]}✱ ✇✐t❤ σ = s−[s] ❛♥❞
k =−1 ♦t❤❡r✇✐s❡✳ ■t ✐s ❛ r❡✢❡①✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠✿
kukWαs, p(RN)=
X
0≤|λ|≤k
k̺α−s+|λ|(lg̺)−1∂λukpLp(RN)
+ X
k+1≤|λ|≤[s]−1
k̺α−s+|λ|∂λukpLp(RN)
1/p
+ X
|λ|=[s]
k∂λukWσ, p
α (RN).
❲❡ ❝❛♥ s✐♠✐❧❛r❧② ❞❡✜♥❡✱ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡r β✱ t❤❡ s♣❛❝❡✿
Wα, βs, p(RN) =
v ∈ D′(RN); (lg̺)βv ∈Wαs, p(RN)
.
❲❡ ❝❛♥ ♣r♦✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s Wα, βs, p(RN)✳ ❲❡
❤❛✈❡ t❤❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ✐♠❜❡❞❞✐♥❣s ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ N/p+α /∈
✻
{σ, . . . , σ+ [s]−1}✿
Wα, βs, p(RN)֒→Wα−1, βs−1, p(RN)֒→ · · ·֒→Wα−[s], βσ, p (RN), Wα, βs, p(RN)֒→Wα+[s]−s, β[s], p (RN)֒→ · · ·֒→Wα−s, β0, p (RN).
❲❤❡♥ N/p+α=j ∈ {σ, . . . , σ+ [s]−1}✱ t❤❡♥ ✇❡ ❤❛✈❡✿
Wα, βs, p ֒→ · · ·֒→Wα−j+1, βs−j+1, p ֒→Wα−j, β−1s−j, p ֒→ · · ·֒→Wα−[s], β−1σ, p ,
Wα, βs, p ֒→Wα+[s]−s, β[s], p ֒→ · · ·֒→Wα−σ−j+1, β[s]−j+1, p ֒→Wα−σ−j, β−1[s]−j, p ֒→ · · ·֒→Wα−s, β−10, p .
■❢ u ✐s ❛ ❢✉♥❝t✐♦♥ ♦♥RN
+✱ ✇❡ ❞❡♥♦t❡ ✐ts tr❛❝❡ ♦❢ ♦r❞❡r j ♦♥ t❤❡ ❤②♣❡r♣❧❛♥❡ Γ
❜②✿
∀j ∈N, γju:x′ ∈RN−1 7−→∂Nj u(x′,0).
▲❡t✬s r❡❝❛❧❧ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛❝❡ ❧❡♠♠❛ ❞✉❡ t♦ ❍❛♥♦✉③❡t ✭s❡❡ ❬✶✻❪✮ ❛♥❞ ❡①t❡♥❞❡❞
❜② ❆♠r♦✉❝❤❡✲◆❡↔❛s♦✈á ✭s❡❡ ❬✺❪✮ t♦ t❤✐s ❝❧❛ss ♦❢ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s✿
▲❡♠♠❛ ✷✳✷✳ ❋♦r ❛♥② ✐♥t❡❣❡r m≥1 ❛♥❞ r❡❛❧ ♥✉♠❜❡r α✱ t❤❡ ♠❛♣♣✐♥❣
γ = (γ0, γ1, . . . , γm−1) : DRN+−→
m−1
Y
j=0
D(RN−1),
❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣✱ st✐❧❧ ❞❡♥♦t❡❞ ❜② γ✱
γ : Wαm, p(RN
+)−→
m−1
Y
j=0
Wαm−j−1/p, p(RN−1).
▼♦r❡♦✈❡r γ ✐s s✉r❥❡❝t✐✈❡ ❛♥❞ Kerγ = W◦m, pα (RN
+)✳
✷✳✹ ❚❤❡ ❙t♦❦❡s s②st❡♠ ✐♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡
❖♥ t❤❡ ❙t♦❦❡s ♣r♦❜❧❡♠ ✐♥ RN
(S) : −∆u+∇π =f ❛♥❞ divu=h ✐♥RN,
❧❡t✬s r❡❝❛❧❧ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧t ♦♥ ✇❤✐❝❤ ✇❡ ❛r❡ ❜❛s❡❞ ✐♥ t❤❡ s❡q✉❡❧✳ ❋✐rst✱
❢♦r ❛♥② k ∈Z✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡
Sk =n(λ, µ)∈Pk× Pk−1∆ ; divλ= 0, −∆λ+∇µ=0o.
❚❤❡♦r❡♠ ✷✳✸ ✭❆❧❧✐♦t✲❆♠r♦✉❝❤❡ ❬✶❪✮✳ ▲❡t ℓ∈Z ❛♥❞ ❛ss✉♠❡ t❤❛t N/p′ ∈ {1, . . . , ℓ}/ ❛♥❞ N/p /∈ {1, . . . ,−ℓ}.
✼
❋♦r ❛♥② (f, g) ∈ W−1, pℓ (RN)×Wℓ0, p(RN) ⊥ S[1+ℓ−N/p′]✱ ♣r♦❜❧❡♠ (S) ❛❞✲
♠✐ts ❛ s♦❧✉t✐♦♥ (u, π)∈ W1, pℓ (RN)×Wℓ0, p(RN)✱ ✉♥✐q✉❡ ✉♣ t♦ ❛♥ ❡❧❡♠❡♥t ♦❢
S[1−ℓ−N/p]✱ ✇✐t❤ t❤❡ ❡st✐♠❛t❡
(λ, µ)∈Sinf[1−ℓ−N/p]
ku+λkW1, p
ℓ (RN)+kπ+µkW0, p
ℓ (RN)
≤ CkfkW−1, p
ℓ (RN)+kgkW0, p
ℓ (RN)
.
❲❡ ❛❧s♦ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❢♦r ♠♦r❡ r❡❣✉❧❛r ❞❛t❛✿
❚❤❡♦r❡♠ ✷✳✹ ✭❆❧❧✐♦t✲❆♠r♦✉❝❤❡ ❬✶❪✮✳ ▲❡t ℓ ∈ Z ❛♥❞ m ≥ 1 ❜❡ t✇♦ ✐♥t❡❣❡rs
❛♥❞ ❛ss✉♠❡ t❤❛t
N/p′ ∈ {1, . . . , ℓ/ + 1} ❛♥❞ N/p /∈ {1, . . . ,−ℓ−m}.
❋♦r ❛♥② (f, g) ∈ Wm−1, pm+ℓ (RN)×Wm+ℓm, p(RN) ⊥ S[1+ℓ−N/p′]✱ ♣r♦❜❧❡♠ (S)
❛❞♠✐ts ❛ s♦❧✉t✐♦♥(u, π)∈Wm+1, pm+ℓ (RN)×Wm+ℓm, p(RN)✱ ✉♥✐q✉❡ ✉♣ t♦ ❛♥ ❡❧❡♠❡♥t
♦❢ S[1−ℓ−N/p]✱ ✇✐t❤ t❤❡ ❡st✐♠❛t❡
(λ, µ)∈Sinf[1−ℓ−N/p]
ku+λkWm+1, p
m+ℓ (RN)+kπ+µkWm, p
m+ℓ(RN)
≤ C
kfkWm−1, p
m+ℓ (RN)+kgkWm, p
m+ℓ(RN)
.
◆♦t❡ t❤❛t ✐❢ ✇❡ s✉♣♣♦s❡ ℓ = 0✱ t❤❡♥ S[1−N/p′] = P[1−N/p′] × {0} ❛♥❞ t❤❡
♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥(f, g)⊥ S[1−N/p′] ✐s ❡q✉✐✈❛❧❡♥t t♦f ⊥P[1−N/p′]✳
✸ ❍♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠s ✇✐t❤ s✐♥❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s
❚❤❡ ✇❛② ✇❡ ✇✐❧❧ t❛❦❡ t♦ s♦❧✈❡ t❤❡ ❙t♦❦❡s s②st❡♠ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢
✈❡r② ✇❡❛❦ s♦❧✉t✐♦♥s t♦ ❤♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠s ✇✐t❤ s✐♥❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐✲
t✐♦♥s✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ❜✐❤❛r♠♦♥✐❝ ♣r♦❜❧❡♠✿ ✜♥❞u∈Wℓ−11, p(RN
+)s♦❧✉t✐♦♥ t♦
t❤❡ ♣r♦❜❧❡♠
(P) : ∆2u= 0 ✐♥ RN+, u=g0 ❛♥❞ ∂Nu=g1 ♦♥ Γ,
✇❤❡r❡g0 ∈Wℓ−11−1/p, p(Γ)❛♥❞ g1 ∈Wℓ−1−1/p, p(Γ)❛r❡ ❣✐✈❡♥✳ ❲❡ ❜❡❣✐♥ t♦ ❞❡✜♥❡ ❢♦r
❛♥② ✐♥t❡❣❡r q✱ t❤❡ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡Bq ❛s ❢♦❧❧♦✇s✿
Bq =nu∈ Pq∆2; u=∂Nu= 0 ♦♥ Γo.
✽
❚❤❡♦r❡♠ ✸✳✶ ✭❆♠r♦✉❝❤❡✲❘❛✉❞✐♥ ❬✽❪✮✳ ▲❡t ℓ∈Z ❛♥❞ ❛ss✉♠❡ t❤❛t
N/p′ ∈ {1, . . . , ℓ/ −1} ❛♥❞ N/p /∈ {1, . . . ,−ℓ+ 1}. ✭✸✳✶✮
❋♦r ❛♥② g0 ∈ Wℓ−11−1/p, p(Γ) ❛♥❞ g1 ∈ Wℓ−1−1/p, p(Γ) s❛t✐s❢②✐♥❣ t❤❡ ❝♦♠♣❛t✐❜✐❧✐t②
❝♦♥❞✐t✐♦♥
∀ϕ∈ B[2+ℓ−N/p′] , hg1,∆ϕiΓ− hg0, ∂N∆ϕiΓ = 0, ✭✸✳✷✮
♣r♦❜❧❡♠ (P) ❛❞♠✐ts ❛ s♦❧✉t✐♦♥ u ∈ Wℓ−11, p(RN
+)✱ ✉♥✐q✉❡ ✉♣ t♦ ❛♥ ❡❧❡♠❡♥t ♦❢
B[2−ℓ−N/p]✱ ✇✐t❤ t❤❡ ❡st✐♠❛t❡
q∈B[2−ℓ−N/p]inf ku+qkW1, p
ℓ−1(RN
+) ≤ C
kg0kW1−1/p, p
ℓ−1 (Γ)+kg1kW−1/p, p ℓ−1 (Γ)
.
❘❡♠❛r❦ ✸✳✷✳ ✐✮ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ℓ= 1✱ ✐❢ 1−N/p′ <0✱ t❤❡♥B[3−N/p′]={0}
❛♥❞ ✐❢ 1−N/p′ ≥0✱ t❤❡♥B[3−N/p′]=B2 =Rx2N✳
✐✐✮ ❲❡ ❛❧s♦ ❡st❛❜❧✐s❤❡❞ ❛ r❡s✉❧t ❢♦r t❤❡ ❧♦✇❡r ❝❛s❡✱ ✇✐t❤u∈Wℓ−20, p(RN
+)✱ ❜✉t ✇❡
❞♦ ♥♦t ✉s❡ ✐t ✐♥ t❤✐s ♣❛♣❡r✳
❲❡ ✇✐❧❧ ❛❧s♦ ♥❡❡❞ ❛ r❡s✉❧t ♦❢ t❤✐s t②♣❡ ❛❜♦✉t t❤❡ ◆❡✉♠❛♥♥ ♣r♦❜❧❡♠ ❢♦r t❤❡
▲❛♣❧❛❝✐❛♥✿ ✜♥❞ u∈Wℓ−20, p(RN
+)s❛t✐s❢②✐♥❣ t❤❡ ♣r♦❜❧❡♠
(Q) : ∆u= 0 ✐♥RN+ ❛♥❞ ∂Nu=g ♦♥ Γ,
✇❤❡r❡ g ∈Wℓ−2−1−1/p, p(Γ)✳
❚❤❡♦r❡♠ ✸✳✸ ✭❆♠r♦✉❝❤❡ ❬✻❪✮✳ ▲❡t ℓ ∈Z ❛♥❞ ❛ss✉♠❡ t❤❛t
N/p′ ∈ {1, . . . , ℓ/ −2} ❛♥❞ N/p /∈ {1, . . . ,−ℓ+ 2}. ✭✸✳✸✮
❋♦r ❛♥② g ∈Wℓ−2−1−1/p, p(Γ) s❛t✐s❢②✐♥❣ t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥
∀ϕ∈ N[ℓ−N/p∆ ′], hg, ϕi
Wℓ−2−1−1/p, p(Γ)×W−ℓ+22−1/p′, p′(Γ)= 0, ✭✸✳✹✮
♣r♦❜❧❡♠ (Q) ❛❞♠✐ts ❛ s♦❧✉t✐♦♥ u ∈ Wℓ−20, p(RN
+)✱ ✉♥✐q✉❡ ✉♣ t♦ ❛♥ ❡❧❡♠❡♥t ♦❢
N[2−ℓ−N/p]∆ ✱ ✇✐t❤ t❤❡ ❡st✐♠❛t❡
q∈N[2−ℓ−N/p]∆inf ku+qkW0, p
ℓ−2(RN
+) ≤ CkgkW−1−1/p, p
ℓ−2 (Γ).
❲✐t❤ t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts ❛s ❢♦r ❚❤❡♦r❡♠ ✸✳✸✱ ✇❡ ❝❛♥ ♣r♦✈❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡
r❡s✉❧t ❢♦r t❤✐s ♣r♦❜❧❡♠✿
❚❤❡♦r❡♠ ✸✳✹✳ ▲❡t ℓ ∈ Z✳ ❯♥❞❡r ❤②♣♦t❤❡s✐s ✭✸✳✶✮✱ ❢♦r ❛♥② g ∈ Wℓ−1−1/p, p(Γ) s❛t✐s❢②✐♥❣ t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✭✸✳✹✮✱ ♣r♦❜❧❡♠ (Q) ❛❞♠✐ts ❛ s♦❧✉t✐♦♥
u∈Wℓ−11, p(RN
+)✱ ✉♥✐q✉❡ ✉♣ t♦ ❛♥ ❡❧❡♠❡♥t ♦❢ N[2−ℓ−N/p]∆ ✱ ✇✐t❤ t❤❡ ❡st✐♠❛t❡
q∈N[2−ℓ−N/p]∆inf ku+qkW1, p
ℓ−1(RN
+) ≤ CkgkW−1/p, p ℓ−1 (Γ).
✾
◆♦✇✱ ✇❡ ✇✐❧❧ ❡st❛❜❧✐s❤ ❛ s✐♠✐❧❛r r❡s✉❧t ❛❜♦✉t t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ❢♦r t❤❡
▲❛♣❧❛❝✐❛♥ ✇✐t❤ ✈❡r② s✐♥❣✉❧❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ✜♥❞u∈Wℓ−2−1, p(RN
+)s❛t✐s✲
❢②✐♥❣ t❤❡ ♣r♦❜❧❡♠
(R) : ∆u= 0 ✐♥RN
+ ❛♥❞ u=g ♦♥Γ,
✇❤❡r❡ g ∈Wℓ−2−1−1/p, p(Γ)✳
❚❤❡♦r❡♠ ✸✳✺✳ ▲❡t ℓ ∈Z✳ ❯♥❞❡r ❤②♣♦t❤❡s✐s ✭✸✳✸✮✱ ❢♦r ❛♥② g ∈Wℓ−2−1−1/p, p(Γ) s❛t✐s❢②✐♥❣ t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥
∀ϕ ∈ A∆[1+ℓ−N/p′], hg, ∂Nϕi
Wℓ−2−1−1/p, p(Γ)×W−ℓ+22−1/p′, p′(Γ) = 0, ✭✸✳✺✮
♣r♦❜❧❡♠ (R) ❛❞♠✐ts ❛ s♦❧✉t✐♦♥ u ∈ Wℓ−2−1, p(RN
+)✱ ✉♥✐q✉❡ ✉♣ t♦ ❛♥ ❡❧❡♠❡♥t ♦❢
A∆[1−ℓ−N/p]✱ ✇✐t❤ t❤❡ ❡st✐♠❛t❡
q∈A∆[1−ℓ−N/p]inf ku+qkW−1, p
ℓ−2 (RN
+) ≤ CkgkW−1−1/p, p
ℓ−2 (Γ).
❋✐rst❧②✱ ✇❡ ♠✉st ❣✐✈❡ ❛ ♠❡❛♥✐♥❣ t♦ tr❛❝❡s ❢♦r ❛ s♣❡❝✐❛❧ ❝❧❛ss ♦❢ ❞✐str✐❜✉t✐♦♥s✳
❲❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡s Yℓ(RN
+) = nv ∈Wℓ−2−1, p(RN
+); ∆v ∈Wℓ+10, p(RN
+)o, Yℓ,1(RN+) = nv ∈Wℓ−2−1, p(RN+); ∆v ∈Wℓ+1,0, p 1(RN+)o.
❚❤❡② ❛r❡ r❡✢❡①✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡✐r ♥❛t✉r❛❧ ♥♦r♠s✿
kvkYℓ(RN
+) =kvkW−1, p
ℓ−2 (RN
+)+k∆vkW0, p
ℓ+1(RN
+), kvkYℓ,1(RN
+) =kvkW−1, p
ℓ−2 (RN
+)+k∆vkW0, p
ℓ+1,1(RN
+).
▲❡♠♠❛ ✸✳✻✳ ▲❡t ℓ ∈ Z✳ ❯♥❞❡r ❤②♣♦t❤❡s✐s ✭✸✳✸✮✱ t❤❡ s♣❛❝❡ DRN+ ✐s ❞❡♥s❡
✐♥ Yℓ(RN
+) ❛♥❞ ✐♥ Yℓ,1(RN
+)✳
Pr♦♦❢✳ ❋♦r ❡✈❡r② ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r ❢♦r♠T ∈Yℓ(RN
+)′✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡
♣❛✐r (f, g)∈ W◦1, p−ℓ+2′ (RN
+)×W−ℓ−10, p′ (RN
+)✱ s✉❝❤ t❤❛t
∀v ∈Yℓ(RN
+), hT, vi=hf, vi◦
W1, p−ℓ+2′ (RN
+)×Wℓ−2−1, p(RN
+)+
Z
RN
+
g∆v❞x. ✭✸✳✻✮
❚❤❛♥❦s t♦ t❤❡ ❍❛❤♥✲❇❛♥❛❝❤ t❤❡♦r❡♠✱ ✐t s✉✣❝❡s t♦ s❤♦✇ t❤❛t ❛♥② T ✇❤✐❝❤
✈❛♥✐s❤❡s ♦♥ DRN+ ✐s ❛❝t✉❛❧❧② ③❡r♦ ♦♥ Yℓ(RN+)✳ ▲❡t✬s s✉♣♣♦s❡ t❤❛t T = 0 ♦♥
DRN+✱ t❤✉s ♦♥ D(RN
+)✳ ❚❤❡♥ ✇❡ ❝❛♥ ❞❡❞✉❝❡ ❢r♦♠ ✭✸✳✻✮ t❤❛t f+ ∆g = 0 ✐♥ RN
+,
✶✵
❤❡♥❝❡ ✇❡ ❤❛✈❡ ∆g ∈ W◦1, p−ℓ+2′ (RN
+)✳ ▲❡t f˜∈ W−ℓ+21, p′ (RN) ❛♥❞ ˜g ∈ W−ℓ−10, p′ (RN)
❜❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ❡①t❡♥s✐♦♥s ❜② 0 ♦❢ f ❛♥❞ g t♦ RN✳ ❚❤❛♥❦s t♦ ✭✸✳✻✮✱ ✐t ✐s
❝❧❡❛r t❤❛tf˜+ ∆˜g = 0 ✐♥RN✱ ❛♥❞ t❤✉s ∆˜g ∈W−ℓ+21, p′ (RN)✳ ◆♦✇✱ t❤❛♥❦s t♦ t❤❡
✐s♦♠♦r♣❤✐s♠ r❡s✉❧ts ❢♦r t❤❡ ▲❛♣❧❛❝❡ ♦♣❡r❛t♦r ✐♥ RN ✭s❡❡ ❬✹❪✮✱ ✇❡ ❝❛♥ ❞❡❞✉❝❡
t❤❛t g˜∈ W−ℓ+23, p′ (RN)✱ ✉♥❞❡r ❤②♣♦t❤❡s✐s ✭✸✳✸✮✳ ❙✐♥❝❡ g˜ ✐s ❛♥ ❡①t❡♥s✐♦♥ ❜② 0✱ ✐t
❢♦❧❧♦✇s t❤❛tg ∈W◦3, p−ℓ+2′ (RN
+)✳ ❚❤❡♥✱ ❜② ❞❡♥s✐t② ♦❢D(RN
+)✐♥W◦3, p−ℓ+2′ (RN
+)✱ t❤❡r❡
❡①✐sts ❛ s❡q✉❡♥❝❡ (ϕk)k∈N ⊂ D(RN+) s✉❝❤ t❤❛t ϕk → g ✐♥ W◦ 3, p−ℓ+2′ (RN+)✳ ❚❤✉s✱
❢♦r ❛♥② v ∈Yℓ(RN
+)✱ ✇❡ ❤❛✈❡
hT, vi=h−∆g, vi◦
W1, p′−ℓ+2(RN
+)×Wℓ−2−1, p(RN
+)+hg,∆vi◦
W3, p′−ℓ+2(RN
+)×Wℓ−2−3, p(RN
+)
= lim
k→∞
h−∆ϕk, vi◦
W1, p−ℓ+2′ (RN
+)×Wℓ−2−1, p(RN
+)+hϕk,∆vi◦
W3, p−ℓ+2′ (RN
+)×Wℓ−2−3, p(RN +)
= lim
k→∞
(
−
Z
RN +
ϕk∆v❞x+
Z
RN +
ϕk∆v❞x
)
= 0,
✐✳❡✳ T ✐s ✐❞❡♥t✐❝❛❧❧② ③❡r♦✳
❋♦r t❤❡ ❞❡♥s✐t② ♦❢DRN+✐♥Yℓ,1(RN
+)✱ t❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ♣r♦♦❢ ❝♦♥❝❡r♥s t❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢❛❝t♦rs ✐♥ t❤❡ ✇❡✐❣❤ts✱ ✇✐t❤ g ∈W−ℓ−1,0, p′ −1(RN
+)✳
❚❤❛♥❦s t♦ t❤✐s ❞❡♥s✐t② ❧❡♠♠❛✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♦❢ tr❛❝❡s✿
▲❡♠♠❛ ✸✳✼✳ ▲❡t ℓ ∈ Z✳ ❯♥❞❡r ❤②♣♦t❤❡s✐s ✭✸✳✸✮✱ t❤❡ tr❛❝❡ ♠❛♣♣✐♥❣ γ0 : DRN+−→ D(RN−1)✱ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣
γ0 :Yℓ(RN
+)−→Wℓ−2−1−1/p, p(Γ) ✐❢ N/p′ ∈ {ℓ/ −1, ℓ, ℓ+ 1},
resp. γ0 :Yℓ,1(RN
+)−→Wℓ−2−1−1/p, p(Γ) ✐❢ N/p′ ∈ {ℓ−1, ℓ, ℓ+ 1}.
▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ●r❡❡♥ ❢♦r♠✉❧❛
∀v ∈Yℓ(RN
+), ∀ϕ∈W−ℓ+23, p′ (RN
+) s✉❝❤ t❤❛t ϕ= ∆ϕ = 0 ♦♥ Γ, h∆v, ϕiW0, p
ℓ+1(RN
+)×W−ℓ−10, p′ (RN
+)− hv,∆ϕi
Wℓ−2−1, p(RN
+)×W◦1, p−ℓ+2′ (RN
+)
=hv, ∂Nϕi
Wℓ−2−1−1/p, p(Γ)×W−ℓ+22−1/p′, p′(Γ)
✭✸✳✼✮
r❡s♣✳ t❤❡ ●r❡❡♥ ❢♦r♠✉❧❛ ❢♦rv ∈Yℓ,1(RN
+)✱ ✇❤❡r❡ t❤❡ ✜rst t❡r♠ ♦❢ t❤❡ ❧❡❢t✲❤❛♥❞
s✐❞❡ ✐s r❡♣❧❛❝❡❞ ❜② h∆v, ϕiW0, p ℓ+1,1(RN
+)×W−ℓ−1,0, p′ −1(RN
+)
✳
Pr♦♦❢✳ ❋✐rst❧②✱ ❧❡t✬s r❡♠❛r❦ t❤❛t ❢♦r ❛♥② ϕ ∈ W−ℓ+23, p′ (RN
+)✱ t❤❡ ❜♦✉♥❞❛r② ❝♦♥✲
❞✐t✐♦♥ ϕ = ∆ϕ = 0 ♦♥Γ ✐s ❡q✉✐✈❛❧❡♥t t♦ ϕ = ∂N2ϕ = 0 ♦♥ Γ✳ ▼♦r❡♦✈❡r✱ ✐❢
✶✶