Equações elípticas semilineares com dependência do gradiente, via passo da montanha

❊q✉❛çõ❡s ❊❧í♣t✐❝❛s ❙❡♠✐❧✐♥❡❛r❡s ❝♦♠

❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❣r❛❞✐❡♥t❡ ♣♦r P❛ss♦ ❞❛

▼♦♥t❛♥❤❛

▲✉✐③ ❋❡r♥❛♥❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❋❛r✐❛

❖r✐❡♥t❛❞♦r✿ P❛✉❧♦ ❈❡s❛r ❈❛rr✐ã♦

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s✱ q✉❡ ♠❡ ❝❛♣❛❝✐t❛ ❡ s✉st❡♥t❛✳ ✧❋❡❧✐③ ♦ ❤♦♠❡♠ q✉❡ ❡♠ ❉❡✉s ❝♦♥✜❛✧✳ ●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ t♦❞♦s q✉❡✱ ❞❡ ✉♠❛ ❢♦r♠❛ ♦✉ ❞❡ ♦✉tr❛✱ ❝♦♥✲ tr✐❜✉ír❛♠ ♣❛r❛ ✈✐❛❜✐❧✐③❛r ❡st❡ tr❛❜❛❧❤♦✳ ❉❡ ✉♠❛ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r P❛✉❧♦ ❈és❛r ❈❛rr✐ã♦ ❛ q✉❡♠ ♠✉✐t♦ ❛❞♠✐r♦ ❝♦♠♦ ♣❡ss♦❛ ❡ ❝♦♠♦ ♣r♦✜ss✐♦♥❛❧✳

❘❡❣✐str♦ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛♦ ❈◆P◗✱ ✐♥st✐t✉✐çã♦ ❞❛ q✉❛❧ ❢✉✐ ❜♦❧s✐st❛ ❞❡ ❛❜r✐❧ ❞❡ ✷✵✵✸ ❛ ❞❡③❡♠❜r♦ ❞❡ ✷✵✵✹ ❝♦♠♦ ❛❧✉♥♦ ❞♦ ❝✉rs♦ ❞❡ ♠❡str❛❞♦✳

❙♦✉ ❣r❛t♦ ❛♦s q✉❡ ❝♦♠♣✉s❡r❛♠ ♠✐♥❤❛ ❜❛♥❝❛✱ ♣r♦❢❡ss♦r❡s ❖❧í♠♣✐♦ ❍✐r♦s❤✐ ▼✐②❛❣❛❦✐ ❡ ❘♦♥❛❧❞♦ ❇r❛s✐❧❡✐r♦ ❆ss✉♥çã♦ ✱ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❝♦♠ s✉❛s ♣r❡s✲ t✐♠♦s❛s s✉❣❡stõ❡s✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ❞❡ q✉❡♠ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❝❛r✐♥❤♦ s❡♠♣r❡ r❡❝❡❜✐❀ ♣❡❧♦ ❛❢❡t♦ ❡ ❝♦❧❛❜♦r❛çã♦ ❞❡ ♠✐♥❤❛s ✐r♠ãs q✉❡ s❡♠♣r❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ❡ à ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛ ♣♦r ♥✉♥❝❛ t❡r ❞❡✐①❛❞♦ q✉❡ ♣❛❧❛✈r❛s ❞❡ â♥✐♠♦ ❡ ✐♥❝❡♥t✐✈♦ ❢❛❧t❛ss❡♠✳

❙✉♠ár✐♦

✵✳✶ ●❧♦ssár✐♦ ❞❡ ◆♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✶ ❘❡❣✉❧❛r✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s ❋r❛❝❛s ✼

✶✳✶ ▼♦t✐✈❛çã♦ ✲

▼ét♦❞♦s ❱❛r✐❛❝✐♦♥❛✐s ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❘❡❣✉❧❛r✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s ❋r❛❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✷✺

✷✳✶ ❋✉♥❝✐♦♥❛✐s ❉✐❢❡r❡♥❝✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❈♦♥str✉çã♦ ❞♦ ❈❛♠♣♦ Ps❡✉❞♦✲●r❛❞✐❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✺ ▲❡♠❛ ❞❡ ❞❡❢♦r♠❛çã♦ ❞❡ ❈❧❛r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✼ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✽ Pr✐♥❝í♣✐♦ ●❡r❛❧ ▼✐♥✐♠❛① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸ ❊q✉❛çõ❡s ❊❧í♣t✐❝❛s ❙❡♠✐❧✐♥❡❛r❡s ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❣r❛❞✐✲

❡♥t❡ ✹✵

❆ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❞❛ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✻✵ ❇ ❈♦♥t✐♥✉✐❞❛❞❡ ❞♦ ❋❧✉①♦ ❡ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ✻✸ ❈ ❖♣❡r❛❞♦r ❞❡ ❙✉♣❡r♣♦s✐çã♦ ❡♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ✻✹ ❉ ❉❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❙♦❜♦❧❡✈ ❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ λ1 ✻✻

❉✳✵✳✶ ❈❛r❛❝t❡r✐③❛çã♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ λ1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✵✳✶

●❧♦ssár✐♦ ❞❡ ◆♦t❛çõ❡s

c ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s C1 ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦

|u_| ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ s❡❥❛ ❡♠ R✱ ♦✉ ❡♠Rn

Lp_{(Ω;R) ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ▲❡❜❡s❣✉❡✲♠❡♥s✉rá✈❡✐s u: Ω→R}

❝♦♠ ♥♦r♠❛✲Lp _{✜♥✐t❛ kuk}

Lp = R

Ω|u|

p_dx1p_{,1≤p <∞,} ❡ q✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r ❛♠❜✐❣✉✐❞❛❞❡✱ ♥♦t❛r❡♠♦s ku_kLp =kuk_p

L∞_{(Ω;R) ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ▲❡❜❡s❣✉❡✲♠❡♥s✉rá✈❡✐s ❡ ❡ss❡♥❝✐❛❧♠❡♥t❡}

❧✐♠✐t❛❞❛s u: Ω_→R❝♦♠ ♥♦r♠❛✲L∞

ku_k∞=essencialmente supx_∈Ω|u(x)| Hm_{(Ω;R) ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ W}m,2 _{❝♦♠ ♥♦r♠❛}

ku_kHm,2 =P_kkDαuk₂✱ ♦♥❞❡

α= (α1, . . . , αn)✱ αi ≥0;|α|=

Pn

i=1αi;Dαu= ∂

α₁₊...+αn

∂x1···∂xn u. ❖✉ ❛✐♥❞❛ ♥♦t❛r❡♠♦s k · k ♣♦r ♥♦r♠❛ ❡♠H1

H1

0 ❢❡❝❤♦ ❞❡ Cc1 ❡♠ H1

dist(A, B) ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B

w_⊂⊂Ω ✐♥❞✐❝❛ q✉❡ ♦ ❢❡❝❤♦ ❞♦ ❛❜❡rt♦w é ❝♦♠♣❛❝t♦ ❡ w_⊂Ω

∇u P❛r❛ u∈W1,p_{(Ω)✱ Ωs✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ R}n_{✱ ❞❡♥♦t❛♠♦s}

∇u=∂u ∂x1, . . . ,

∂u ∂xn

τhu(x) ✐♥❞✐❝❛ u(x+h)

suppf ✐♥❞✐❝❛ ♦ s✉♣♦rt❡ ❞❛ ❢✉♥çã♦ f E′ _{❊s♣❛ç♦ ❞✉❛❧ ❞❡ ❊}

JacH ❏❛❝♦❜✐❛♥♦ ❞♦ ♦♣❡r❛❞♦r H

Γ =∂Ω ❋r♦♥t❡✐r❛ ❞♦ ❛❜❡rt♦ Ω (r.s), r, s_∈Rn Pr♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ Rn

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ♥ós ❛♣r❡s❡♥t❛♠♦s ✉♠❛ té❝♥✐❝❛ ♥♦✈❛ ♥❛ ❛❜♦r❞❛❣❡♠ ❞♦ ♣r♦✲ ❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ ♥ã♦ ❧✐♥❡❛r ❞♦ ❣r❛❞✐❡♥t❡ ❞❛ s♦❧✉çã♦ ❡♠ ❞✐♠❡♥sã♦ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ três ✭✈❡❥❛ ❉✳ ❋✐❣✉❡✐r❡❞♦✱ ▼✳ ●✐r❛r❞✐✱ ▼✳ ▼❛t③❡✉ ✲ ❬✾❪✮✱ ♦✉ s❡❥❛✿

−△u = f(x, u,∇u) ❡♠ Ω

u = 0 s♦❜r❡ Γ =∂Ω ✭✶✮

♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ s✉❛✈❡ ❡ ❧✐♠✐t❛❞♦ ❡♠ Rn, n≥3✳

❉✐③❡♠♦s q✉❡ ❛ ❛❜♦r❞❛❣❡♠ q✉❡ ❛♣r❡s❡♥t❛♠♦s é ✉♠❛ té❝♥✐❝❛ ♥♦✈❛ ♣♦✐s✱ ❛té ❡♥tã♦✱ ❛s ❢♦r♠❛s ❛♣r❡s❡♥t❛❞❛s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✭1✮ ♥ã♦ ❡♥✈♦❧✈✐❛♠ ❛

t❡♦r✐❛ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ✭✈❡r✿ ❏✳ ❇✳ ▼✳ ❳❛✈✐❡r ✲ [✷✵]✱ ❳✳ ❲❛♥❣ &❨✳ ❉❛♥❣ ✲ [✷✶]✱ ❩✳ ❨❛♥ [✷✷]✮✱ ✉♠❛ ✈❡③ q✉❡ ✭1✮ ♥ã♦ é ✈❛r✐❛❝✐♦♥❛❧✳

▼❛s ❝♦♠♦ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ ♥ã♦ ✈❛r✐❛❝✐♦♥❛❧ ❝♦♠ ❛ t❡♦r✐❛ ❞❡ P♦♥t♦s ❈rít✐❝♦s❄ ❘❡❛❧♠❡♥t❡ ♥❡st❡ ❝❛s♦✱ ♥ã♦ é ♣♦ssí✈❡❧ ❛♣❧✐❝❛r♠♦s ❡st❛ t❡♦r✐❛ ❞✐r❡t❛✲ ♠❡♥t❡✳ ▼❛s ❝♦♥t♦r♥❛♠♦s ❡st❡ ❡♠♣❡❝✐❧❤♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❛♦ ♣r♦❜❧❡♠❛ ✭1✮✱

❛ss♦❝✐❛♠♦s ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s❡♠✐❧✐♥❡❛r❡s s❡♠ ❞❡♣❡♥❞ê♥✲ ❝✐❛ ❞♦ ❣r❛❞✐❡♥t❡ ❞❛ s♦❧✉çã♦❀ ✐st♦ é✱ ♣❛r❛ ❝❛❞❛ w _∈ H1

0✱ ♥ós ❝♦♥s✐❞❡r❛♠♦s ♦

♣r♦❜❧❡♠❛

−△u = f(x, u,_∇w) ❡♠ Ω

u = 0 s♦❜r❡ Γ =∂Ω . ✭✷✮

❆❣♦r❛ ♦ ♣r♦❜❧❡♠❛ ✭2✮✱ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❤✐♣ót❡s❡s q✉❡ ♠❡♥❝✐♦♥❛r❡♠♦s

❛❜❛✐①♦✱ é ✈❛r✐❛❝✐♦♥❛❧ ❡ t❡♠ ♣♦r ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ Iw : H01(Ω) → R ❞❛❞♦

♣♦r

Iw(v) =

1 2

Z

Ω|∇ v|2₋

Z

Ω

F(x, v,∇w).

❈♦♥❥✉♥t♦ ❞❡ ❤✐♣ót❡s❡s✿

(f0) f : Ω_×R_×Rn _{é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ ❧♦❝❛❧♠❡♥t❡ ❞❡ ▲✐♣s❝❤✐t③✳}

(f1) limt→0 f(x,t,ξ_t ) = 0 ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ x∈Ω❡ ξ ∈Rn✳

(f2) ❊①✐st❡♠ ❝♦♥st❛♥t❡s a1 >0❡ p_∈ 1,n+2

n₋2

t❛✐s q✉❡ |f(x, t, ξ)| ≤a1(1 +|t|p

) ∀x∈Ω, t∈R, ξ ∈Rn.

(f3) ❊①✐st❡♠ ❝♦♥st❛♥t❡s θ >2 ❡t0 >0t❛✐s q✉❡

0< θF(x, t, ξ)_≤tf(x, t, ξ) _∀x_∈Ω,_|t_{| ≥}t0, ξ_∈Rn

♦♥❞❡

F(x, t, ξ) =

Z t

0

f(x, s, ξ)ds.

(f4) ❊①✐st❡♠ ❝♦♥st❛♥t❡s a2✱ a3 t❛✐s q✉❡ F(x, t, ξ)≥a2|t|θ

−a3 ∀x∈Ω, t∈R, ξ∈Rn.

(f5✮ f s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ❞❡ ▲✐♣s❝❤✐t③ ❧♦❝❛✐s

|f(x, t′, ξ)−f(x, t′′, ξ)| ≤Lρ1|t′−t′′|

∀x_∈Ω, _|t′_{| ≤ρ}

1, |t′′| ≤ρ1, |ξ| ≤ρ2,

|f(x, t, ξ′)₋f(x, t, ξ′′)_{| ≤}Lρ2|ξ′−ξ′′|

∀x_∈Ω, _|t_{| ≤}ρ1, |ξ′| ≤ρ2, |ξ′′| ≤ρ2,

♦♥❞❡ Lρ1✱Lρ2 sã♦ t❛✐s q✉❡ q✉❡ ✈❡r✐✜❝❛♠ ❛ r❡❧❛çã♦

λ−₁1Lρ1 +λ

−12

1 Lρ2 <1, ✭✸✮

♦♥❞❡ λ1 é ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞❡ −△ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ Ω ❡ ρ2, ρ2 sã♦

❞❛❞♦s ♥♦ ❧❡♠❛ 3.8✱ ❡ ❞❡♣❡♥❞❡♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡ p✱ n✱ θ✱ a1✱ a2 ❡ a3

❞❛❞♦s ♥❛s ❤✐♣ót❡s❡s ❛♥t❡r✐♦r❡s✳

❖❜s✳❆ ❝♦♥❞✐çã♦ ✭3✮ é ❝♦♠♦ ❞❛❞❛ ❡♠ ▼✳ ●✐r❛r❞✐ &▼✳ ▼❛t③❡✉ ✲ [✶✸]✳

❆s ❤✐♣ót❡s❡s (f0) ❛ (f4) ❣❛r❛♥t❡♠ q✉❡ ♦ ♣r♦❜❧❡♠❛ ✭2✮ ♣♦ss✉✐ s♦❧✉çã♦✳

❆❣r❡❣❛♥❞♦ ❛ ❤✐♣ót❡s❡ (f5)✱ ❝♦♥s❡❣✉✐r❡♠♦s ♠♦str❛r q✉❡ ♦ ♣r♦❜❧❡♠❛ (1) ♣♦s✲

s✉✐ s♦❧✉çã♦✳

❖ ❛rt✐❣♦ ❝❡♥tr❛❧ q✉❡ ✐r❡♠♦s tr❛❜❛❧❤❛r ♥❡st❛ ❞✐ss❡rt❛çã♦ ✭❉✳ ❞❡ ❋✐❣✉❡✐❡❞♦✱ ▼✳ ●✐r❛r❞✐ & ▼✳ ▼❛t③❡✉ ✲ [✾]✮✱ ♥♦s ❞✐③ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ✉t✐❧✐③❛r té❝♥✐❝❛s

❞♦ ❈á❧❝✉❧♦ ❞❛s ❱❛r✐❛çõ❡s ♣❛r❛ r❡s♦❧✈❡r♠♦s ♣r♦❜❧❡♠❛s ❞❡ ❊q✉❛çõ❡s ❊❧í♣t✐❝❛s ❙❡♠✐❧✐♥❡❛r❡s ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❣r❛❞✐❡♥t❡ ✈✐❛ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ q✉❡r❡♠♦s ✐❧✉str❛r ❝♦♠♦ té❝♥✐❝❛s ❞♦ ❈á❧❝✉❧♦ ❞❛s ❱❛r✐❛çõ❡s ♣♦❞❡♠ ✧❢❛❝✐❧✐t❛r✧ ❛ ♣r♦❝✉r❛ ❞❡ s♦❧✉çõ❡s ❡♠ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✱ ❡ ♠♦t✐✈❛r ♦ ✉s♦ ❞❡ t❛✐s ♠ét♦❞♦s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❝❡♥tr❛❧✳ ❉❡✜♥✐♠♦s ❛ ♥♦çã♦ ❞❡ s♦❧✉çã♦ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦ ❡ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ♥♦s ❣❛r❛♥t❡♠ ❛ r❡❣✉❧❛r✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s ❢r❛❝❛s ✭r❡❢❡rê♥❝✐❛s ✉t✐❧✐③❛❞❛s [✷]✱ [✻] ❡ [✶✷]✮✳ ❊♥✉♥❝✐❛♠♦s ❡ ♣r♦✈❛♠♦s ♦ ❚❡♦r❡♠❛ ✶✳✷✳✶ ❡ ❝✐t❛♠♦s✱ s❡♠ ❞❡♠♦♥str❛r✱

♦s ❚❡♦r❡♠❛s ✶✳✺✳✶ ❡ ✶✳✺✳✷✳ ❙♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ♦s ❚❡♦r❡♠❛s ✶✳✷✳✶✱ ✶✳✺✳✶ ❡ ✶✳✺✳✷ ♥♦s ❞✐③❡♠ q✉❡ ✉♠❛ s♦❧✉çã♦ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦ ♣❡rt❡♥❝❡ ❛ H2_{✱ W}2,p _{❡ C}2

r❡s♣❡❝t✐✈❛♠❡♥t❡✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ♣r♦✈❛♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ❞❡ ❆♠❜r♦s❡tt✐ ❡ ❘❛❜✐♥♦✇✐t③✱ q✉❡ ♥♦s ❛✉①✐❧✐♦✉ ♥❛ ❜✉s❝❛ ♣♦r ♣♦♥t♦s ❝rít✐❝♦s ❞♦ ❢✉♥❝✐♦♥❛❧ ❛s✲ s♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭2✮✳ Pr♦✈❛♠♦s t❛♠❜é♠ ♦ ▲❡♠❛ ❞❛ ❉❡❢♦r♠❛çã♦ ❞❡ ❈❧❛r❦

❡ ♦s ❚❡♦r❡♠❛s ❞❡ ❊❦❡❧❛♥❞✱ ❇ré③✐s✲◆✐r❡♥❜❡r❣✱ ❙❤✉❥✐❡ ▲✐ ❡ ♦ Pr✐♥❝í♣✐♦ ●❡r❛❧ ▼✐♥✐♠❛① ✭[✶✽]✱ [✶✾]✮✳ ❱❡r❡♠♦s q✉❡ ♦ ✐♥❣r❡❞✐❡♥t❡ ❝❤❛✈❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦

❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ é ♦ ▲❡♠❛ ❞❛ ❞❡❢♦r♠❛çã♦ ❞❡ ❈❧❛r❦✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❡♥✉♥❝✐❛♠♦s ❡ ♣r♦✈❛♠♦s ♦s s❡❣✉✐♥t❡s ❚❡♦r❡♠❛s✿

❚❡♦r❡♠❛ ✵✳✶✳✶✳ ❙✉♣♦♥❤❛ (f0),(f1),(f2),(f3),(f4) ✈á❧✐❞♦s✳ ❊♥tã♦ ❡①✐st❡♠

❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s c1 ❡ c2 t❛✐s q✉❡✱ ♣❛r❛ ❝❛❞❛ w ∈ H1

0(Ω)✱ ♦ ♣r♦❜❧❡♠❛ ✭✷✮

t❡♠ ✉♠❛ s♦❧✉çã♦ uw t❛❧ q✉❡ c1 ≤ kuwk ≤ c2✳ ❆❧é♠ ❞✐ss♦✱ s♦❜ ❛s ❤✐♣ót❡s❡s

❛❝✐♠❛✱ ✭✷✮ t❡♠ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡ ✉♠❛ ♥❡❣❛t✐✈❛✳ ❆ ❞❡♠♦♥str❛çã♦ ❢♦✐ ❢❡✐t❛ ❡♠ ✈ár✐❛s ❡t❛♣❛s✳

❚❡♦r❡♠❛ ✵✳✶✳✷✳ ❙✉♣♦♥❤❛ ❛s ❝♦♥❞✐çõ❡s (f0)✱. . .✱(f5) ✈á❧✐❞❛s✱ ❡♥tã♦ ♦ ♣r♦❜❧❡✲

♠❛ ✭3.1✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡ ✉♠❛ ♥❡❣❛t✐✈❛ ♥♦ s❡♥t✐❞♦ ❝❧áss✐❝♦✳

❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❚❡♦r❡♠❛ ❝♦♥s✐st❡✱ ❜❛s✐❝❛♠❡♥t❡✱ ❡♠ ❡①tr❛✐r ❞❛ ❢❛♠í❧✐❛ ❞❡ s♦❧✉çõ❡s ❛ss♦❝✐❛❞❛s ❛♦ ♣r♦❜❧❡♠❛ ✭2✮✱ ♣♦r ♠ét♦❞♦s ✐t❡r❛t✐✈♦s✱ ✉♠❛

s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠ H1

0 ❝✉❥♦ ❧✐♠✐t❡ é ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛(1)✳

❈❛♣ít✉❧♦ ✶

❘❡❣✉❧❛r✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s ❋r❛❝❛s

✶✳✶ ▼♦t✐✈❛çã♦ ✲

▼ét♦❞♦s ❱❛r✐❛❝✐♦♥❛✐s ❡♠ ❊q✉❛çõ❡s ❉✐❢❡✲

r❡♥❝✐❛✐s

❖ ▼ét♦❞♦ ❉✐r❡t♦ ❞♦ ❈á❧❝✉❧♦ ❞❛s ❱❛r✐❛çõ❡s ❝♦♥s✐st❡ ♥❛ ♦❜t❡♥çã♦ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s✱ ♣❛r❛ ✉♠ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❞❡ ♠♦❞♦ ♥❛t✉r❛❧ ❛♦ ♣r♦❜❧❡♠❛ ❞✐❢❡r✲ ❡♥❝✐❛❧✳ ❊ss❛ ✐❞é✐❛ ❞❡ tr❛t❛r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❛tr❛✈és ❞❡ ✉♠ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♣❛r❡❝❡ ❡♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳■❳✱ ❞❡ ♠♦❞♦ ❡①♣❧í❝✐t♦ ❝♦♠ ❉✐r✐❝❤✲ ❧❡t ❡ ❘✐❡♠❛♥♥✳ ▼♦str❛r❡♠♦s✱ ♥❡st❡ ❝❛♣ít✉❧♦✱ ❛ ❛♣❧✐❝❛çã♦ ❞❡ss❛s té❝♥✐❝❛s ♥❛ r❡s♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t ❤♦♠♦❣ê♥❡♦✳

❋♦r♠✉❧❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥t♦r♥♦ ❡❧í♣t✐❝♦ ❉❛❞♦ x_∈Rn_{✱ ❡s❝r❡✈❡r❡♠♦s}

x= (x′, xn)♦♥❞❡ x′ ∈Rn−1, x′ = (x1, x2, . . . , xn₋1) ❡ xn∈R,

❡ ♣♦r❡♠♦s

|x′_|=

n−1

X

i=1 x2i

!1 2

.

◆♦t❛r❡♠♦s ❛✐♥❞❛

Rn

+ ={x= (x′, xn);xn>0},

Q=_{x= (x′_{, xn);|x}′_{|<1 ❡|xn|<1},}

Q+ =QTRn+, Q0 =_{x= (x′_{, x}

n);|x′|<1 ❡ xn= 0}.

❉❡✜♥✐çã♦ ✶✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❛❜❡rt♦ Ω⊂ Rn é ❞❡ ❝❧❛ss❡ Cm_{✱ ♦♥❞❡ m ≥1}

❡ ✐♥t❡✐r♦✱ s❡ ♣❛r❛ ❝❛❞❛ x_∈Γ =∂Ω❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ✭❡♠Rn_{✮ ❞❡ x ❡}

✉♠❛ ❜✐❥❡çã♦ H :Q_→U t❛❧ q✉❡

H ∈Cm(Q)✱ H−1 ∈Cm(U)✱ H(Q+) =U ∩Ω✱ H(Q0) =U ∩Γ.

❖ ❛❜❡rt♦ Ω é ❞❡ ❝❧❛ss❡ C∞ _{s❡ ❢♦r ❞❡ ❝❧❛ss❡ C}m _{♣❛r❛ t♦❞♦ ♠✳}

Pr♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t ❤♦♠♦❣ê♥❡♦ ✲ ❙❡❥❛Ω∈Rn✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦

❞❡ ❝❧❛ss❡ C1_{❀ ❜✉s❝❛♠♦s ✉♠❛ s♦❧✉çã♦u: Ω→Rq✉❡ ✈❡r✐✜❝❛}

−△u+u =f ❡♠ Ω

u = 0 s♦❜r❡ Γ =∂Ω . ✭✶✳✶✮

△u=

n

X

i=1 ∂2u ∂x2

i

é ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ u ❡ f é ✉♠❛ ❢✉♥çã♦ ❞❛❞❛ ❡♠ Ω✳ ❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦

u= 0 s♦❜r❡ Γ✭❢r♦♥t❡✐r❛✮ s❡ ❝❤❛♠❛ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ✭❤♦♠♦❣ê♥❡❛✮✳

❯♠❛ s♦❧✉çã♦ ❝❧áss✐❝❛ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ é ✉♠❛ ❢✉♥çã♦ u ∈ C2_{(Ω) q✉❡}

s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❡♠ ✭✶✳✶✮✳

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✶✳✶✮ ♣♦r v _∈ C1

c(Ω)✱ ❡ ✐♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s✱

♦❜t❡♠♦s

Z

Ω∇

u_∇v+

Z

Ω uv =

Z

Ω

f v, _∀v _∈C_c1(Ω).

❊ s❡❣✉❡ ♣♦r ❞❡♥s✐❞❛❞❡

Z

Ω∇

u_∇v+

Z

Ω uv =

Z

Ω

f v, _∀v _∈H₀1(Ω). ✭✶✳✷✮

❆ ❡①♣r❡ssã♦ ✭✶✳✷✮ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿ ✉♠❛ ❢✉♥çã♦u_∈H1

0 é ✉♠❛

s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✶✳✶✮ s❡ u s❛t✐s❢❛③ à r❡❧❛çã♦ ✭✶✳✷✮✳

❆❧❣♦ s✉r♣r❡❡♥❞❡♥t❡ q✉❡ ✈❡r❡♠♦s é q✉❡ ❛ ♣r♦❝✉r❛ ♣♦r ✉♠❛ s♦❧✉çã♦ ❝❧áss✐❝❛ s❡ ✧r❡❞✉③✧ ❛ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛✳

❚♦❞❛ s♦❧✉çã♦ ❝❧áss✐❝❛ é s♦❧✉çã♦ ❢r❛❝❛

◗✉❡ ✉♠❛ s♦❧✉çã♦ ❝❧áss✐❝❛ ❞❡ ✭✶✳✶✮ s❛t✐s❢❛③ ✭✶✳✷✮ ♥ós ❥á s❛❜❡♠♦s✳ ❇❛st❛✱ ❡♥tã♦✱ ♥♦t❛r♠♦s q✉❡ u∈H1

0✳ ❈♦♠ ❡❢❡✐t♦✱u∈H1(Ω)∩C(Ω) ❡ u= 0 s♦❜r❡ Γ =∂Ω

❡ ♣♦rt❛♥t♦ u_∈H1 0✳

❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦❧✉çã♦ ❢r❛❝❛ P❛r❛ ✈❡r✐✜❝❛r♠♦s ❡st❡ ❢❛t♦✱ t♦♠❡♠♦s

(u, v) =

Z

Ω

(_∇u._∇v+uv)

♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtH =H1

0✱ ❡ ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ϕ∈H′ ϕ :v 7→

Z

f v.

❖ ❚❡♦r❡♠❛ ❞❛ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③ ♥♦s ❞✐③ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ u ∈ H

t❛❧ q✉❡

< ϕ, v >=ϕ(v) = (u, v) _∀v _∈H,

♦✉ s❡❥❛ _Z

Ω

(∇u.∇v+uv) =

Z

Ω

f v ∀v ∈H.

❘❡❣✉❧❛r✐❞❛❞❡ ❞❛ s♦❧✉çã♦ ❢r❛❝❛

❊st❡ é ✉♠ ♣♦♥t♦ ❞❡❧✐❝❛❞♦ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❡ ✉♠ t❛♥t♦ q✉❛♥t♦ s✉r♣r❡❡♥❞❡♥t❡✱ ❡ ♣♦rt❛♥t♦ ✈❛♠♦s ❛❜♦r❞á✲❧♦ ❝♦♠ ❞❡t❛❧❤❡ ♥❛ s❡çã♦ ✶✳✷✳

❘❡❝✉♣❡r❛çã♦ ❞❛ s♦❧✉çã♦ ❝❧áss✐❝❛ ❙✉♣♦♥❞♦ q✉❡ ❛ s♦❧✉çã♦u∈H1

0 ❞❡ ✭✶✳✶✮ ♣❡rt❡♥❝❡ ❛C2(Ω).❊♥tã♦u= 0 s♦❜r❡

Γ✳ P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s

Z

Ω

(−△u+u)v =

Z

Ω

f v ∀v ∈Cc1(Ω)

❡ ♣♦rt❛♥t♦(_−△u+u) =f ♣❛r❛ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❡♠Ω❥á q✉❡C1

c(Ω)é ❞❡♥s♦

❡♠ L2_{(Ω)✳ ❈♦♠♦ u ∈ C}2_{(Ω)✱ s❡❣✉❡ q✉❡ (−△u+u) = f ❡♠ t♦❞♦ ♣♦♥t♦ ❞❡}

Ω. ❆ss✐♠✱ u é s♦❧✉çã♦ ♥♦ s❡♥t✐❞♦ ❝❧áss✐❝♦ ❞❡ ✭✶✳✶✮✳

✶✳✷ ❘❡❣✉❧❛r✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s ❋r❛❝❛s

❚❡♦r❡♠❛ ✶✳✷✳✶✳ ❬❘❡❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t❪ ❙❡❥❛ Ω ✉♠

❛❜❡rt♦ ❞❡ ❝❧❛ss❡ C2 _{❝♦♠ Γ ✭❢r♦♥t❡✐r❛✮ ❧✐♠✐t❛❞❛✳ ❙❡❥❛♠ f ∈ L}2_{(Ω) ❡ u ∈} H1

0(Ω) t❛✐s q✉❡

Z

Ω∇

u_∇ϕ+

Z

Ω uϕ=

Z

Ω

f ϕ _∀ϕ_∈H₀1(Ω). ✭✶✳✸✮

❊♥tã♦✱u∈H2_{(Ω) ❡||u||}

H2 ≤C||f||₂ ♦♥❞❡C é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ só ❞❡♣❡♥❞❡

❞❡ Ω. ❆❧é♠ ❞✐ss♦✱ s❡ Ω é ❞❡ ❝❧❛ss❡ Cm+2 _{❡ s❡ f ∈H}m_{✱ ❡♥tã♦}

u∈Hm+2(Ω) ❝♦♠ ||u||Hm+2 ≤C||f||_m.

❉❡♠♦♥str❛çã♦✿ ❋❛r❡♠♦s ❡st❛ ❞❡♠♦♥str❛çã♦ ❡♠ três ❡t❛♣❛s✳ ❊t❛♣❛ ❆✿ Ω =Rn_.

◆♦t❛çã♦ ✶✳ ❉❛❞♦ ❤ ∈R✱ h₆= 0 ♣♦r❡♠♦s

Dhu=

1

|h|(τhu−u), ♦✉ s❡❥❛✱ (Dhu)(x) =

u(x+h)₋u(x)

|h| .

❋❛ç❛♠♦s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s ❛♥t❡s ❞❡ ♣r♦ss❡❣✉✐r♠♦s✳ ❉❛❞♦su, v❢✉♥çõ❡s

q✉❛✐sq✉❡r ❞❡✜♥✐❞❛s ❞❡ Rn _{❡♠ R✳}

Z

Rn

(Dhu)vdx =

Z

Rn

u(x+h)−u(x)

|h_| v(x)dx

=

Z

Rn

u(x+h)

|h_| v(x)dx−

Z

Rn

u(x)

|h_| v(x)

=

Z

Rn

u(y)

|h| v(y−h)dx−

Z

Rn

u(x)

|h| v(x)

=

Z

Rn

v(x−h)−v(x)

|h_| x(x)dx=

Z

Rn

u(D₋hv)dx. ✭✶✳✹✮

❈♦♠♦ ♥❡st❛ ❡t❛♣❛ ❆ ❞❛ ❞❡♠♦♥str❛çã♦Ω = Rn✱ ♥♦s ♣❡r♠✐t✐r❡♠♦s ❝♦❧♦❝❛r

❛♣❡♥❛s R ♣❛r❛ ✐♥❞✐❝❛r R_Rn✳

❙❡❥❛u✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞♦ ♣r♦❜❧❡♠❛ ✧✐♥✐❝✐❛❧✧ ✭✶✳✶✮✳ ❈♦♠♦u∈H1_(Rn_{) =}

H1

0(Rn)✱ ♥♦t❡ q✉❡ ❡♠ ✭✶✳✸✮ ♣♦❞❡♠♦s t♦♠❛r ϕ =D−h(Dhu)✳ ❆ss✐♠✱ ❞❡ ✭✶✳✹✮

♦❜t❡♠♦s _Z

|∇Dhu|2+

Z

|Dhu|2 =

Z

f D₋h(Dhu).

❊ ❡♥tã♦

||Dhu||2H1 =

Z

f D₋h(Dhu)≤ kfk2kD−h(Dhu)k2. ✭✶✳✺✮

❉❛ Pr♦♣♦s✐çã♦ ❆✳✶✱ t❡♠♦s q✉❡||τhv−v||L2_(ω) ≤ |h|||∇v||_L2_(Rn₎ ∀ω ⊂⊂Rn✱ ❡ ♣♦rt❛♥t♦

♦✉ ❛✐♥❞❛

||D₋hv||L2_(Rn₎ ≤ ||∇v||_L2_(Rn). ✭✶✳✻✮ ❉❡ ✭✶✳✺✮ ❡ ✭✶✳✻✮✱

||Dhu||2H1 ≤ kfk2||∇(Dhu)k2 ≤ kfk2kDhukH1,

❡ ❡♠ ♣❛rt✐❝✉❧❛r

Dh

∂u ∂xj

2

≤ kDhuk2 ≤ ||Dhu||H1 ≤ kfk2.

❆ss✐♠✱

τh

∂u ∂xj −

∂u ∂xj

2

≤ |h|kfk2,

❡ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✶ ♦❜t❡♠♦s q✉❡

∂u ∂xj ∈

H1 _∀j = 1,2, . . . , n

❡✱ ♣♦rt❛♥t♦✱ u_∈H2_.

◗✉❡r❡♠♦s ♦❜s❡r✈❛r ❛❣♦r❛ q✉❡ s❡ f ∈ Hn_{✱ ❡♥tã♦✱ u ∈ H}n+2_{✳ ❋❛r❡♠♦s ♦}

❝❛s♦

f _∈H1 _⇒u_∈H3,

❡ ♦ ❝❛s♦ ❣❡r❛❧ s❡❣✉❡ ✐♥❞✉t✐✈❛♠❡♥t❡✳

◆♦t❡♠♦s ♣♦r Du q✉❛❧q✉❡r ❞❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s _∂x∂u

j✱ 1 ≤ j ≤ n✳ P❛r❛

f _∈ H1_{✱ ✈✐♠♦s q✉❡ u ∈ H}2_{✱ ♦✉ s❡❥❛✱ Du ∈ H}1_{✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡} Du∈H2_{✳ ◆♦t❡ q✉❡ s❡ ♠♦str❛r♠♦s}

Z

∇(Du)_∇ϕ+

Z

(Du)ϕ=

Z

(Df)ϕ _∀ϕ _∈H₀1, ✭✶✳✼✮

❝❛í♠♦s ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❡ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✳ ❙❡❥❛ ❞❛❞♦ϕ ∈C∞

c (Rn)✳ ❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✳✸✮ ♣♦r Dϕ♦❜t❡♠♦s

Z

∇u_∇(Dϕ) +

Z

u(Dϕ) =

Z

f(Dϕ)

❡✱ ♣♦rt❛♥t♦✱ _Z

∇(Du)_∇ϕ+

Z

(Du)ϕ =

Z

(Df)ϕ.

❊ ❝♦♠♦ C∞

c é ❞❡♥s♦ ❡♠ H1(Rn)✱ s❡❣✉❡ ✭✶✳✼✮✳

❊t❛♣❛ ❇✿Ω =Rn₊.

❉❡✜♥✐çã♦ ✷✳ ❉✐r❡♠♦s q✉❡ h é ♣❛r❛❧❡❧♦ à ❢r♦♥t❡✐r❛ s❡ h _∈ Rn−1 _{× {0}✱ ❡}

❡s❝r❡✈❡r❡♠♦s h//Γ✳

❖❜s✳ ❉❛❞♦u _∈H1

0(Ω)✱ s❡ h//Γ✱ ❡♥tã♦✱ τhu∈ H01(Ω)✳ ❉❡ ❢❛t♦✱ t♦♠❡♠♦s un ∈Cc1(Ω) q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ u ❡♠ H1(Ω)✳ ◆♦t❡ q✉❡ τhun ∈ Cc1(Ω)✱ ❡ q✉❡

τhun ❝♦♥✈❡r❣❡ ♣❛r❛ τhu ❡♠ H1(Ω)✳

❉❛❞♦h//Γ✱ ♣♦♥❞♦ ϕ=D₋h(Dhu) ❡♠ ✭✶✳✸✮ t❡♠♦s

Z

Ω|∇

Dhu|2 +

Z

Ω

Dhu=

Z

Ω

f D₋h(Dhu),

♦✉ ❛✐♥❞❛ ♣♦r ❍ö❧❞❡r

||Dhu||2H1 ≤ kfk2kD−h(Dhu)k2. ✭✶✳✽✮

▲❡♠❛ ✶✳✸✳ ||Dhu||L2_(Ω) ≤ k∇uk_L2_(Ω) ∀u∈H1(Ω),∀h//Γ.

❉❡♠♦♥str❛çã♦✿

❈♦♠❡❝❡♠♦s s✉♣♦♥❞♦ q✉❡ u_∈C1

c(Rn)✳ ❙❡❥❛ h∈Rn ❡ ❞❡✜♥❛♠♦s

v(t) =u(x+th), t∈R.

❊♥tã♦✱ v′_{(t) = h.∇u(x+th) ❡}

u(x+h)−u(x) =v(1)−v(0) =

Z 1 0

v′(t)dt =

Z 1 0

h.∇u(x+th)dt.

P♦rt❛♥t♦✱

|τhu(x)−u(x)|2 ≤ |h|2

Z 1 0

|∇u(x+th)|2_dt.

❚♦♠❛♥❞♦ h//Γ✱ ❝♦♠♦ Ω +th = Ω✱ t❡♠♦s

Z

Ω|

τhu(x)−u(x)|2dx ≤ |h|2

Z

Ω

Z 1 0 |∇

u(x+th)_|2_dtdx

= _|h_|2

Z 1 0

Z

Ω|∇

u(x+th)_|2dxdt

= |h|2

Z 1 0

Z

Ω+th

|∇u(y)|2_dydt

= _|h_|2

Z 1 0

Z

Ω|∇

u(y)_|2dydt.

❊ ❛ss✐♠

||τhu−u||2L2_(Ω) ≤ |h|2

Z

Ω|∇

u_|2. ✭✶✳✾✮

▲♦❣♦✱

||Dhu||L2_(Ω) ≤ k∇uk_L2_(Ω) ∀u∈C_c1(Rn),∀h//Γ.

❈♦♠♦ Rn₊ é ❡♠ ♣❛rt✐❝✉❧❛r C1_{✱ ♣♦r ❞❡♥s✐❞❛❞❡}✶ _t❡♠♦s

||Dhu||L2_(Ω) ≤ k∇uk_L2_(Ω) ∀u∈H1(Ω),∀h//Γ.

P♦r ✭✶✳✽✮ ❡ ♦ ▲❡♠❛ 1.3✱ t❡♠♦s

kDhuk2H1 ≤ kfk2k∇(Dhu)k2 ≤ kfk2k∇(Dhu)kH1,

❞♦♥❞❡

kDhukH1 ≤ kfk₂ ∀ h//Γ. ✭✶✳✶✵✮

❉❛❞♦s1_≤j _≤n✱1_≤k _≤n₋1✱h =_|h_|ek ❡ ϕ∈Cc∞(Ω)✱ t❡♠♦s

−

Z

uD₋h

∂ϕ ∂xi

=

Z

Dh

∂u ∂xi

ϕ

q✉❡ ♣♦r ❍ö❧❞❡r ❡ ✭✶✳✶✵✮ s❡❣✉❡

✶_❙❡ΩéC1 _{❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡sC}∞

c (R

n₎r❡str✐t❛s ❛_Ωé ❞❡♥s♦ ❡♠

H1_(Ω)✳

Z

uD₋h

∂ϕ ∂xi ≤ Dh ∂u ∂xi 2

kϕ_{k2 ≤ k}DhukH1kϕk₂ ≤ ||f||2||ϕ||2.

P❛ss❛♥❞♦ ♦ ❧✐♠✐t❡ |h_{| →}0♦❜t❡♠♦s

Z u ∂ 2_ϕ

∂xi∂xk

≤ kfk2kϕk2 ∀1≤i≤n, ∀ 1≤k≤n−1. ✭✶✳✶✶✮

❆✜r♠❛çã♦✿ Z u∂ 2_ϕ ∂x2 n

≤ kfk2kϕk2 ∀ϕ ∈Cc∞(Ω).

❉❡ ❢❛t♦✱ ❞❡ ✭✶✳✸✮ t❡♠♦s

Z

∇u_∇ϕ =

Z

(f ₋u)ϕ,

♦✉ ❛✐♥❞❛✱ Z ∂x ∂xn ∂ϕ ∂xn −

n−1

X i=1 Z ∂u ∂xi ∂ϕ ∂xi ≤ n X i=1 Z ∂u ∂xi ∂ϕ ∂xi = Z

(f₋u)ϕ

❡✱ ♣♦rt❛♥t♦✱ Z u∂ 2_ϕ ∂x2 n ≤ Z

(f ₋u)ϕ

+

n−1

X i=1 Z u∂ 2_ϕ ∂x2 i .

❖❜t❡♠♦s ❞❡ ✭✶✳✶✶✮ ❡ ❞♦ ❢❛t♦ ku_k2

H1 ≤ kfk2kukH1 q✉❡

Z u∂ 2_ϕ ∂x2 n

≤ kfk2kϕk2 ∀ϕ ∈Cc∞(Ω). ✭✶✳✶✷✮

❆ss✐♠✱ ❞❡ ✭✶✳✶✶✮ ❡ ✭✶✳✶✷✮ t❡♠♦s

Z u ∂ 2_ϕ

∂xi∂xk

≤ kfk2kϕk2 ∀ 1≤i≤n, ∀ 1≤k≤n.

❊ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✶ s❡❣✉❡ q✉❡ u_∈H2_(Ω)✳

❋✐♥❛❧♠❡♥t❡✱ ❞❡♠♦♥str❡♠♦s q✉❡ s❡ f ∈ Hm_{(Ω)✱ ❡♥tã♦✱ u ∈ H}m+2_(Ω)✳

◆♦✈❛♠❡♥t❡✱ ❢❛r❡♠♦s ♦ ❝❛s♦

f ∈H1 ⇒u∈H3,

❡ ♦ ❝❛s♦ ❣❡r❛❧ s❡❣✉❡ ✐♥❞✉t✐✈❛♠❡♥t❡✳

❉❡♥♦t❡♠♦s ♣♦r Du q✉❛❧q✉❡r ❞❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s t❛♥❣❡♥❝✐❛✐s Du =

∂u

∂xj, 1≤j ≤n−1✳

▲❡♠❛ ✶✳✹✳ ❙❡❥❛ u_∈H2_(Ω)∩H1

0(Ω) q✉❡ ✈❡r✐✜❝❛ ✭✶✳✸✮✳ ❊♥tã♦ Du ∈H01(Ω)

❡ _Z

∇(Du)∇ϕ+

Z

(Du)ϕ=

Z

(Df)ϕ ∀ϕ∈H₀1. ✭✶✳✶✸✮

❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ ϕ _∈C∞

c ✱ s❡ s✉❜st✐t✉✐r♠♦sDϕ ❡♠ ✭✶✳✸✮ ♥♦ ❧✉❣❛r

❞❡ ϕ✱ ♦❜t❡♠♦s _Z Ω∇

Du_∇ϕ+

Z

Ω

Duϕ=

Z

Ω Df ϕ,

❡ ♣♦r ❞❡♥s✐❞❛❞❡ s❡❣✉❡ ✭✶✳✶✸✮✳

❘❡st❛✲♥♦s ♠♦str❛r q✉❡ Du _∈ H1

0. ❚♦♠❡♠♦s h = |h|ej, 1 ≤ j ≤ n −1❀

❡♥tã♦ Dhu∈H₀1 ✭H₀1 é ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❧❛çõ❡s t❛♥❣❡♥❝✐❛✐s✮✳

P❡❧♦ ▲❡♠❛ ✶✳✸ t❡♠♦s

kDhukH1 ≤ kfk₂.

❈♦♠♦ H1

0 é ❍✐❧❜❡rt ❡ Dhnu✱ hn → 0 é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ♣❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡❝❡ssár✐♦✱ Dhnu ⇀ g ✭❢r❛❝❛♠❡♥t❡✮✷✳ ▲♦❣♦✱ ♣❡❧❛ ✐♥✲ ❝❧✉sã♦ ❞❡ ❙♦❜♦❧❡✈ ✭❚❡♦✳ D.0.1✮✱Dhnu ⇀ g ❡♠ L

2_.

P❛r❛ϕ ∈C∞

c (Ω) t❡♠♦s✱

Z

(Dhu)ϕ=

Z

u(D₋hϕ).

❋❛③❡♥❞♦ hn→0

Z

gϕ=

Z

u∂ϕ ∂xj ∀

ϕ_∈C_c∞(Ω), ♦✉ s❡❥❛✱ ∂u ∂xj

=g _∈H₀1.

✷_{❉❛❞❛(xn)∈E✱xn⇀ x⇔< f, xn>→< f, x >∀f ∈E}′

.

❙❡❣✉❡ ❞❡

Z

u∂ 2_ϕ

∂x2

n

≤

Z

(f ₋u)ϕ

+

n−1

X

i=1

Z

u∂ 2_ϕ

∂x2

i

,

❡ ❞❛ Pr♦♣♦s✐çã♦ ✭❆✳✷✮ q✉❡

∂u ∂xn ∈

H₀1.

P❡❧♦ ♣r♦❝❡ss♦ ❥á ❡①✐❜✐❞♦✱

∂u ∂xj ∈

H01 ∀1≤j ≤n

❡✱ ♣♦rt❛♥t♦✱ u∈H3_(Ω)✳

❊t❛♣❛ ❈✿ ❈❛s♦ ❣❡r❛❧✳

❋✐①❡ ✉♠❛ ❢✉♥çã♦ζ ∈C∞

c (Rn) t❛❧ q✉❡ 0≤ζ ≤1❡

ζ(x)

1 se _|x_{| ≤}1 0 se |x| ≥2 .

❊ ❞❡✜♥❛ ❛ s❡q✉ê♥❝✐❛

ζn(x) =ζ

_x

n

♣❛r❛ ♥❂✶✱✷✱✳ ✳ ✳

❊st❛ s❡q✉ê♥❝✐❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ s❡q✉ê♥❝✐❛ ❞❡ tr✉♥❝❛♠❡♥t♦✳ ❙❡ ❝♦♠♣r♦✈❛✱ s❡♠ ❞✐✜❝✉❧❞❛❞❡✱ q✉❡ ❞❛❞♦ u_∈H1_{✱ supp(ζ}

nu)é ❝♦♠♣❛❝t♦ ♣❛r❛ ❝❛❞❛ n ∈N❡

ζnu ❝♦♥✈❡r❣❡ ❡♠ H2 ♣❛r❛ u✳

❙❡❥❛ u ∈ H1

0(Ω)✱ s♦❧✉çã♦ ❞❡ ✭✶✳✸✮✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ s❡ f ∈ L2(Ω)✱

❡♥tã♦✱ u_∈ H2_{(Ω)✳ P❛r❛ s✐♠♣❧✐✜❝❛r✱ ✈❛♠♦s s✉♣♦r Ω ❧✐♠✐t❛❞♦✱ ❝❛s♦ ❝♦♥trár✐♦}

t♦♠❛r❡♠♦s ζnu q✉❡ é s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦

−△(ζnu) +ζnu=ζnf −2∇ζn∇u−(△ζn)u def

≡ g.

❈❧❛r❛♠❡♥t❡ ζnu ♣❡rt❡♥❝❡ ❛ H₀1(Ω) ❡ t❡♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ✭❡ ❛ss✐♠ ♣♦❞❡✲

♠♦s s✉♣♦r ❞♦♠í♥✐♦ Ω′ _{❧✐♠✐t❛❞♦✮✳ ◆♦t❡ q✉❡ g ∈ L}2_(Ω′_{)✳ ❙❡ ♣r♦✈❛r♠♦s q✉❡}

ζnu ∈ H2(Ω′)✱ t❡r❡♠♦s q✉❡ ζnu∈ H2(Ω′)∩H2(Ω′)⊂ H2(Ω) ❡ ♣❡❧❛ ❝♦♠♣❧❡✲

t✉❞❡ ❞❡ H2_{(Ω)✱ t❡r❡♠♦s q✉❡ u∈H}2_(Ω)✳

P❛r❛ ❝♦♠♣r♦✈❛r q✉❡ ζnu é s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦

−△(ζnu) +ζnu=g,

❜❛st❛ ♥♦t❛r♠♦s q✉❡ ❞❛❞♦ w_∈H1 0(Ω)✱

Z

Ω

[−△(ζnu)w+ζnuw−ζnf w+ 2∇ζn∇uw+ (△ζn)uw] = 0

s❡ ❡ s♦♠❡♥t❡ s❡

Z

Ω

∇(ζnu)∇w+

Z

Ω

ζnuw−

Z

Ω

ζnf w+ 2

Z

Ω

∇ζn∇uw−

Z

Ω

∇ζn∇(uw) = 0

s❡ ❡ s♦♠❡♥t❡ s❡

Z

Ω∇

u_∇(wζn) +

Z

Ω

uwζn =

Z

Ω f wζn.

❈♦♠♦ ✉ s❛t✐s❢❛③ ✭✶✳✸✮✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ t♦♠❛♥❞♦v =wζn.

▲♦❣♦✱ s✉♣♦♥❞♦Ω❧✐♠✐t❛❞♦ t❡♠♦s q✉❡ ∂Ω = Γé ✉♠ ❝♦♠♣❛❝t♦ ❞❡ Rn✳ ❙❡❥❛

Sk

i=1Ui ⊃Γ✉♠❛ ❝♦❜❡rt✉r❛ ✜♥✐t❛ ❞❡Γt❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ i❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦

Hi :Q→Ui ❝♦♠

Hi ∈C1(Q)✱ Hi−1 ∈C1(U)✱ Hi(Q+) =Ui∩Ω✱ Hi(Q0) = Ui ∩Γ.

❊♥tã♦✱ t♦♠❡♠♦s ✉♠❛ ♣❛rt✐çã♦ ❞❛ ✉♥✐❞❛❞❡✱ θ0, θ1, θ2, . . . , θk ∈ C∞(Rn)✱ s✉✲

❜♦r❞✐♥❛❞❛ ❛ ❡ss❛ ❝♦❜❡rt✉r❛✱ ✐st♦ é✱ suppθi ⊂Ui ✱suppθ0 ⊂Ω✱ ❝♦♠

✭✐✮ 0_≤θi ≤1 ∀i= 0, . . . , k ❡

Pk

i=1 = 1 ❡♠ Ω

✭✐✐✮ supp θi é ❝♦♠♣❛❝t♦✳

❚❘❆❚❆▼❊◆❚❖ ◆❖ ■◆❚❊❘■❖❘ ✲ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡θ0 ∈H2(Ω)

❈♦♠♦ θ0 ∈ C∞

c (Ω)✱ ❛ ❢✉♥çã♦ ❡st❡♥❞✐❞❛ ♣♦r ③❡r♦ ❢♦r❛ ❞❡ Ω ♣❡rt❡♥❝❡ ❛

H1_(Rn_{)✳ ◆♦t❡ q✉❡ θ}

0u é s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦

−△(θ0u) +θ0u=θ0f−2∇θ0∇u−(△θ0)udef≡ g.

◆♦t❡ t❛♠❜é♠ q✉❡g _∈L2_{✱ ❞❡ ❢❛t♦}

kg_k2 ≤2kθ0k2kfk2+ 2k∇θ0k2k∇uk2+k△θ0k2kuk2 ≤c(kfk2+kuk2).

❈♦♠♦ ♥♦ ❝❛s♦ ❆✱ ♦❜t❡♠♦s q✉❡ θ0u _∈ H2_(Rn_{)✱ ❧♦❣♦✱ θ0u ∈ H}2_{(Ω)✳ ❊}

❛✐♥❞❛✱ ❝♦♠♦ θ0u é s♦❧✉çã♦ ❢r❛❝❛ ❞❡

−△(θ0u) +θ0u=g,

t♦♠❛♥❞♦ v =uθ0✱ t❡♠♦s

kθ0u_k2 =

Z

Ω∇

(θ0u)_∇(θ0u) +

Z

Ω

θ0uθ0u=

Z

Ω

gθ0u_{≤ k}g_k2kθ0ukH1.

❉❛í✱

kθ0uk ≤c(kfk2+kuk2)≤

∼

c _kfk2,

❥á q✉❡ ku_kH1 ≤ kfk₂✳

❊❙❚■▼❆❚■❱❆ ◆❆ ❋❘❖◆❚❊■❘❆ ✲ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ θiu ∈ H2(Ω)

♣❛r❛ ❝❛❞❛ i= 1,2, . . . , k.

❋✐①❡♠♦si_{∈ {}1,2, . . . , k}✳ ❈♦♠♦ H é ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡Q ❡Ui✱ ♣♦❞❡♠♦s

❡s❝r❡✈❡r

x=H(y) ❡H−1(x) = J(x) ∀x∈Ui.

◆♦t❡ q✉❡ v =θiu∈H01(Ω∩Ui)✱ ❡ q✉❡ v =θiu é s♦❧✉çã♦ ❢r❛❝❛ ❡♠ Ω∩Ui

❞❛ ❡q✉❛çã♦

−△v =θif−θiu−2∇θi∇u−(△θi)u def

≡ g.

❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r ♦❜t❡♠♦s kg_k2 ≤ckfk2.

❆ss✐♠✱ ✭❧❡♠❜r❛♥❞♦ q✉❡ v =θiu✮ t❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

Z

Ω∩Ui

∇v_∇ϕdx=

Z

Ω∩Ui

gϕdx _∀ϕ _∈H1

0(Ω∩Ui). ✭✶✳✶✹✮

❚r❛♥s♣♦rt❛r❡♠♦s v_|Ω∩Ui ♣❛r❛ Q+✱ ♣❛r❛ ✉t✐❧✐③❛r♠♦s ❛ ♥♦çã♦ ❞❡ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s t❛♥❣❡♥❝✐❛✐s✳ ❆ss✐♠ ♣♦♥❤❛♠♦s

w(y) =v(H(y)) ♣❛r❛y _∈Q+ w(J(x)) =v(x) ♣❛r❛x_∈Ω_∩Ui.

▲❡♠❛ ✶✳✺✳ ❈♦♠ ❛s ♥♦t❛çõ❡s ❛♥t❡r✐♦r❡s t❡♠♦s✿ w∈H1

0(Q+) ❡

n X k,l=1 Z Q+ akl ∂w ∂yk ∂ψ ∂yl dy= Z Q+ ∼

g ψdy _∀ψ _∈(Q+), ✭✶✳✶✺✮

♦♥❞❡ ∼g_{= (g ◦H)|JacH| ∈ L}2_{(Q+) ❡ ❛s ❢✉♥çõ❡s a}

kl ∈ C1(Q+) ✈❡r✐✜❝❛♠ ❛s

❝♦♥❞✐çõ❡s ❞❡ ❡❧✐♣t✐❝✐❞❛❞❡ ✭✶✳✶✻✮

n

X

i,j=1

ai,j(x)ξiξj ≥α|ξ|2 ∀x∈Ω, ∀ξ∈Rn✱ ♣❛r❛ ❛❧❣✉♠ α >0. ✭✶✳✶✻✮

❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ ψ _∈ H1

0(Q+)✱ ♣♦♥❤❛♠♦s ϕ(x) = ψ(J(x)), x ∈ (Ω∩Ui)✳ ❊♥tã♦

❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ψn∈Cc1(Q+) t❛❧ q✉❡

ψn(J(x))→ψ(J(x)) =ϕ.

❚♦♠❛♥❞♦ ϕn(x) = ψn ◦J(x) q✉❡ ❝❧❛r❛♠❡♥t❡ ♣❡rt❡♥❝❡ ❛ Cc1(Ω∩Ui), s❡❣✉❡

q✉❡ ϕ_∈H1

0(Ω∩Ui)✳ P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s q✉❡

∂v ∂xj =X k ∂w ∂yk ∂Jk ∂xj ✱ ∂ϕ ∂xj =X l ∂ψ ∂yl ∂Jl ∂xj . ❆ss✐♠✱ Z

Ω∩Ui

∇v∇ϕdx =

Z

Ω∩Ui

X j ∂v ∂xj ∂ϕ ∂xj = Z

Ω∩Ui

X j " X k ∂w ∂yk ∂Jk ∂xj ! X l ∂ψ ∂yl ∂Jl ∂xj !# dx = Z Q+ X j,k,l ∂Jk ∂xj ∂Jl ∂xj ∂w ∂yk ∂ψ ∂yl|

JacH_|dy

= Z Q+ X k,l X j ∂Jk ∂xj ∂Jl

∂xj|

JacH_| ! ∂w ∂yk ∂ψ ∂yl dy. ❖✉ s❡❥❛✱ Z

Ω∩Ui

∇v_∇ϕdx=

♦♥❞❡

akl =

X

j

∂Jk

∂xj

∂Jl

∂xj|

JacH|.

❈♦♠♦ J _∈C2_(U

i)✱ t❡♠♦s q✉❡ akl ∈C1(Q+)✳

❱❡r✐✜q✉❡♠♦s ❛s ❝♦♥❞✐çõ❡s ❞❡ ❡❧✐♣t✐❝✐❞❛❞❡ ❞❡ akl. ❉❛❞♦ ξ ∈ Rn✱ ξ =

(ξ1, . . . , ξn)✱

X

k,l

aklξkξl =

X k,l X j ∂Jk ∂xj ∂Jl

∂xj|

JacH_|

!

ξkξl

= |JacH|X

j X k,l ∂Jk ∂xj ξk ∂Jl ∂xj ξl !

= _|JacH_|X

j X k ∂Jk ∂xj ξk !2

≥α_|ξ_|2_,

❝♦♠ α >0✱ ✈✐st♦ q✉❡JacJ ❡JacH sã♦ ♥ã♦ s✐♥❣✉❧❛r❡s✳ ❊st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧✲

❞❛❞❡ ❡stá ♣r♦✈❛❞❛ ♥❛ Pr♦♣♦s✐çã♦ ❆✳✸✳ P♦r ♦✉tr♦ ❧❛❞♦✱

Z

Ω∩Ui

gϕdx=

Z

Q+

(g◦H)ψ|JacH|dy. ✭✶✳✶✽✮

❉❡ ✭✶✳✶✹✮✱ ✭✶✳✶✼✮ ❡ ✭✶✳✶✽✮ t❡♠♦s

Z Q+ X k,l akl ∂w ∂yk ∂ψ ∂yl dy = Z Q+

(g◦H)ψ|JacH|dy.

❉❡♠♦♥str❡♠♦s ❛❣♦r❛ q✉❡ w _∈ H2_{(Q) ❡ kwk}

H2 ≤ k

∼

g _k2. ✭■st♦ ✐♠♣❧✐❝❛rá

q✉❡θiu♣❡rt❡♥❝❡ ❛H2(Ω∩Ui)❝♦♠kθiukH2 ≤ckfk₂ ❡✱ ♣♦rt❛♥t♦✱θ_iu♣❡rt❡♥❝❡

❛ H2_(Ω)✮✳

❈♦♠♦ ♥♦ ❝❛s♦ ❇✱ ♦♥❞❡Ω =Rn✱ ✉t✐❧✐③❛r❡♠♦s tr❛♥s❧❛çõ❡s t❛♥❣❡♥❝✐❛✐s✳ ❊♠

✭✶✳✶✺✮✱ t♦♠❡♠♦s ψ = D₋h(Dhw) ❝♦♠ h//Q0 ❡ |h| s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦

♣❛r❛ q✉❡ ψ _∈ H1

0(Q+)✳ ◆♦t❡ q✉❡ s❡ v = θiu t❡♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ ❡♥tã♦✱

suppw ⊂ {(x′_{, xn);|x}′_{|<1−delta ❡ 0< xn<1−δ}✳}

❖❜t❡♠♦s ❡♥tã♦ X k,l Z Q+ Dh akl ∂w ∂yk ∂ ∂yl

(Dhw) =

Z

Q+

∼

g D₋h(Dhw). ✭✶✳✶✾✮

▼❛s

Z

Q+

∼

g D₋h(Dhw) ≤ k

∼

g _{k2kD−h(Dhw)k2 ✭✶✳✷✵✮}

≤ k∼g _{k2k∇(Dhw)k2.}

P♦r ♦✉tr♦ ❧❛❞♦✱

Dh akl ∂w ∂yk

(y) =

akl(y+h)

∂w ∂yk

(y+h)₋akl

∂w ∂yk

1

|h_|

=akl(y+h)

∂ ∂yk

Dhw(y) + (Dhakl(y))

∂w ∂yk

(y)

❡✱ ♣♦rt❛♥t♦✱ X k,l Z Q+ Dh akl ∂w ∂yk ∂ ∂yl

(Dhw) =

X

k,l

Z

Q+

akl(y+h)

∂ ∂yk

Dhw(y)

∂ ∂yl

(Dhw)dy

+ X

k,l

Z

Q+

(Dhakl(y))

∂w ∂yk

(y) ∂

∂yl

(Dhw)dy.

❆ss✐♠✱ X k,l Z Q+ Dh akl ∂w ∂yk ∂ ∂yl

(Dhw) ≥ αk∇Dhwk22 ✭✶✳✷✶✮

− CkwkH1k∇D_hwk2.

❉❡ ✭✶✳✶✾✮✱ ✭✶✳✷✵✮ ❡ ✭✶✳✷✶✮

k ∼g _k2k∇Dhwk2 ≥αk∇Dhwk22−CkwkH1k∇D_hwk₂,

♦✉ ❛✐♥❞❛

αk∇Dhwk2 ≤ k

∼

g _k2+CkwkH1.

❊ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r

k∇Dhwk2 ≤

∼

C k∼g _{k2 ✭✶✳✷✷✮}

♣♦✐s ❞❡ ✭✶✳✶✺✮ s❡❣✉❡ q✉❡

α_kw_k2_H1 ≤

n X k,l=1 Z Q+ akl ∂w ∂yk ∂w ∂yl dy = Z Q+ ∼

g wdy _{≤ k}∼g _k2kw_k2_H1.

❙❡❥❛♠ 1_≤k_≤n✱1_≤l _≤n₋1✱ h=_|h_|ek ❡ ψ ∈Cc∞(Ω)✳

Z

Q+

Dh

∂w ∂yk

ψ =−

Z

Q+

(Dhw)

∂ψ ∂yk

❡ ❞❡ (1.22)

Z

Q+

Dh

∂w ∂yk

ψ _{≤ k∇}Dhwk2kψk2 ≤

∼

C k ∼g _k2kψ_k2.

❖✉ s❡❥❛ Z Q+

(Dhw)

∂ψ ∂yk ≤ ∼

C k∼g _k2kψ_k2.

P❛ss❛♥❞♦ ♦ ❧✐♠✐t❡✿ h_→0

Z Q+ ∂w ∂yl ∂ψ ∂yk ≤ ∼

C k∼g _k2kψ_k2. ✭✶✳✷✸✮

P❛r❛ ❝♦♥❝❧✉✐r♠♦s q✉❡ w ∈H2_{(Q+) ✭❡ kwk}

H2 ≤ck

∼

g _{k2✮✱ ❜❛st❛ ♠♦str❛r✲}

♠♦s q✉❡ Z Q+ ∂w ∂yn ∂ψ ∂yn ≤ ∼

Ck ∼g _k2kψ_k2. ✭✶✳✷✹✮

❊♠ ✭✶✳✶✺✮ s✉❜st✐t✉✐♥❞♦ ψ ♣♦r 1

annψ✱ ❝♦♠ ann ≥α > 0✱ ❡ ann ∈ C

1_(Q +)✱ s❡❣✉❡ q✉❡ Z Q+ ∼ g ψ ann dy = n X k,l=1 Z Q+ akl ∂w ∂yk ∂ ∂yl ψ ann dy = n X k,l=1 Z Q+ akl ∂w ∂yk ∂ψ ∂yl 1

ann −

■s♦❧❛♥❞♦ ♦ t❡r♠♦ q✉❡ q✉❡r❡♠♦s ❡st✐♠❛r ♦❜t❡♠♦s Z Q+ ∂w ∂yn ∂ψ ∂yn

dy = X

(k,l)6=(n,n)

Z

Q+

aklψ

a2

nn

∂w ∂yk

∂ann

∂yl −

X

(k,l)6=(n,n)

Z Q+ akl ann ∂w ∂yk ∂ψ yl + Z Q+ ∼ g ψ ann dy+ Z Q+ ∂w ∂yn ∂ann ∂yn ψ ann dy.

❘❡❛❣r✉♣❛♥❞♦ ♦s t❡r♠♦s ❝♦♥s❡❣✉✐♠♦s q✉❡

Z Q+ ∂w ∂yn ∂ψ ∂yn

dy = X

(k,l)6=(n,n)

Z Q+ ∂w ∂yk ψ ann ∂akl

∂yl −

X

(k,l)6=(n,n)

Z Q+ ∂ ∂yl akl ann ψ ∂w ∂yk + Z Q+ ∼ g ψ ann dy+ Z Q+ ∂w ∂yn ∂ann ∂yn ψ ann dy.

❆ss✐♠✱ ❝♦♠❜✐♥❛♥❞♦ ✭✶✳✷✸✮ ❡ ✭✶✳✷✺✮✱ ♦♥❞❡ ❡♠ ✭✶✳✷✸✮ é ♥❡❝❡ssár✐♦ t♦♠❛r♠♦s

akl

annψ ♥♦ ❧✉❣❛r ❞❡ ψ✱ ♦❜t❡♠♦s

Z Q+ ∂w ∂yn ∂ψ ∂yn

≤c(kwkH1 +k

∼

g _k2)_kψ_k2 ∀ ψ ∈Cc1(Q+),

❞♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ✭✶✳✷✹✮✳

❋✐♥❛❧♠❡♥t❡✱ s❡f _∈Hm_{(Ω)✱ ❡♥tã♦✱ θu∈H}m+2_{(Ω) ♣❛r❛ t♦❞♦θ ∈ C}∞

c (Ω)✳

P❛r❛ ✈❡r✐✜❝❛r♠♦s ❡st❡ ❢❛t♦✱ ❜❛st❛ r❡♣❡t✐r♠♦s ❛s ❡st✐♠❛t✐✈❛s ❢❡✐t❛s ♥♦ ✐♥t❡r✐♦r ❞❡ Ω♣❛r❛ ♦ ❝❛s♦

f _∈H1 _⇒θu_∈H3,

❡ ♦ ❝❛s♦ ❣❡r❛❧ s❡❣✉❡ ✐♥❞✉t✐✈❛♠❡♥t❡✳

❊♥✉♥❝✐❛r❡♠♦s ❛❣♦r❛✱ s❡♠ ❞❡♠♦♥str❛r✱ ♠❛✐s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ r❡✲ ❣✉❧❛r✐❞❛❞❡ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♥♦ ❈❛♣ít✉❧♦ 3✳

❚❡♦r❡♠❛ ✶✳✺✳✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ h _∈Lp_{(Ω)✱ 1< p < ∞ ❡ q✉❡ u∈ H}1 0(Ω)

s❡❥❛ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞♦ ♣r♦❜❧❡♠❛

−△u = h(x) ❡♠ Ω

u = 0 s♦❜r❡ Γ =∂Ω ✭✶✳✷✺✮

❊♥tã♦✱ u_∈W2,p_(Ω)✳

❉❡♠♦♥str❛çã♦✿

◆♦t❡ q✉❡ ♦ ❝❛s♦p= 2 ♣r♦✈❛♠♦s ♥♦ ❚❡♦r❡♠❛1.22✳ ❖ ❝❛s♦ ❣❡r❛❧✱ ✈✐❞❡ ❆❣♠♦♥ [✷]✱ ❚❤✳✽✳✷

❚❡♦r❡♠❛ ✶✳✺✳✷✳ ❙❡❥❛ 0< α≤1 ❡ s✉♣♦♥❤❛♠♦s q✉❡ u∈Cα_(Ω)∩H1

0(Ω) s❡❥❛

✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞♦ ♣r♦❜❧❡♠❛ ✭1.25✮ ❝♦♠ h_∈Cα_{(Ω)✳ ❊♥tã♦ u∈C}2,α_(Ω)✳

❉❡♠♦♥str❛çã♦✿

❱✐❞❡ ●✐❧❜❛r❣ & ❚r✉❞✐♥❣❡r[✶✷]✱ ❚❤✳✻✳✶✹

❈❛♣ít✉❧♦ ✷

❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛

❖ ❢♦❝♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ♣❛r❛ ❢✉♥❝✐♦♥❛✐s✳ Pr♦✈❛r❡✲ ♠♦s ❛q✉✐ ❛ ✈❡rsã♦ ✉s✉❛❧ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡✱ ♣❛r❛ t❛❧✱ ♦ ✐♥❣r❡❞✐❡♥t❡ ❝❤❛✈❡ ♥❡st❛ ❞❡♠♦♥str❛çã♦ é ♦ ▲❡♠❛ ❞❡ ❉❡❢♦r♠❛çã♦ ❞❡ ❈❧❛r❦✳ ❊ ✈❡r❡♠♦s ❝♦♠♦ ❡st❡ ✐♠♣♦rt❛♥t❡ ❚❡♦r❡♠❛ ♥♦s ❛✉①✐❧✐❛ ♥❛ ❜✉s❝❛ ❞❡ ♣♦♥t♦s ❝rít✐✲ ❝♦s ♣❛r❛ ❢✉♥❝✐♦♥❛✐s✳ Pr♦✈❛r❡♠♦s t❛♠❜é♠ ♦ ▲❡♠❛ ❞❛ ❉❡❢♦r♠❛çã♦ ❞❡ ❈❧❛r❦ ❡ ♦s ❚❡♦r❡♠❛s ❞❡ ❊❦❡❧❛♥❞✱ ❇ré③✐s✲◆✐r❡♥❜❡r❣✱ ❙❤✉❥✐❡ ▲✐ ❡ ♦ Pr✐♥❝í♣✐♦ ●❡r❛❧ ▼✐♥✐♠❛① ✭[✶✽]✱[✶✾]✮✳

✷✳✶ ❋✉♥❝✐♦♥❛✐s ❉✐❢❡r❡♥❝✐á✈❡✐s

◆❡st❡ ❝❛♣ít✉❧♦✱ k · k ✐♥❞✐❝❛rá ❛ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦ ♠étr✐❝♦ ❡♠ q✉❡stã♦✳ ❱❛♠♦s r❡❧❡♠❜r❛r ❛❧❣✉♠❛s ♥♦çõ❡s ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✸✳ ❙❡❥❛ ϕ:U _→R ♦♥❞❡U é ✉♠ ❛❜❡rt♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ X✳ ❖ ❢✉♥❝✐♦♥❛❧ ϕ é ●❛t❡❛✉① ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ u ∈ U s❡ ❡①✐st❡ f ∈ X′_{✱ t❛❧}

q✉❡ ♣❛r❛ t♦❞♦ h_∈X✱

lim

t_→0

1

t[ϕ(u+th)−ϕ(u)− hf, thi] = 0.

❙❡ ♦ ❧✐♠✐t❡ ❛❝✐♠❛ ❡①✐st✐r✱ ❡❧❡ é ú♥✐❝♦ ❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① ❡♠ u s❡rá

❞❡♥♦t❛❞❛ ♣♦r ϕ′_{(u)✱ ❡ ❞❛❞❛ ♣♦r}

hϕ′(u), h_i:= lim

t→0

1

t[ϕ(u+th)−ϕ(u)].

❖ ❢✉♥❝✐♦♥❛❧ ϕ t❡♠ ❞❡r✐✈❛❞❛ ❛ ❋ré❝❤❡t f ∈X′ _{❡♠ u s❡}

lim

h→0

1

kh_k[ϕ(u+h)−ϕ(u)− hf, hi] = 0.

❖ ❢✉♥❝✐♦♥❛❧ ϕ ♣❡rt❡♥❝❡ ❛ C1_{(U,R) s❡ ϕ ♣♦ss✉✐ ❞❡r✐✈❛❞❛ ❛ ❋ré❝❤❡t ❡ ❡st❛ é}

❝♦♥tí♥✉❛ ❡♠ U✳

❙❡ X é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ ϕ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❛ ●❛t❡❛✉① ❡♠ u∈ U✱ ♦

❣r❛❞✐❡♥t❡ ❞❡ ϕ ❡♠ u é ❞❡✜♥✐❞♦ ♣♦r

(∇ϕ(u), h) := hϕ′(u), hi.

❖❜s✳❖ ❢✉♥❝✐♦♥❛❧ ❞✐❢❡r❡♥❝✐á✈❡❧ ❛ ❋ré❝❤❡t é ❞✐❢❡r❡♥❝✐á✈❡❧ ❛ ●❛t❡❛✉①✳

Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡ ϕ t❡♠ ❞❡r✐✈❛❞❛ ●❛t❡❛✉① ❝♦♥tí♥✉❛ ❡♠ U✱ ❡♥tã♦✱ ϕ _∈ C1_(U,R).

❉❡♠♦♥str❛çã♦✿

❉❛❞♦su0 ∈U✱ h∈X ❡ ϕ′ _{❛ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉①✳ ❉❡✜♥❛F :X→R ♣♦♥❞♦}

F(u) = ϕ(u)_{− h}ϕ′_(u

0), u−u0i.

P❡❧♦ t❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱

|F(u)−F(u0)| = |ϕ(u)− hϕ′(u0), u−u0i −ϕ(u0)| ✭✷✳✶✮ ≤ sup

0≤θ≤1k

ϕ′(u0+θ(u−u0))−ϕ′(u0)kku−u0k.

❈♦♠♦ϕt❡♠ ❞❡r✐✈❛❞❛ ●❛t❡❛✉① ❝♦♥tí♥✉❛ ❡♠U✱ ❡♥tã♦✱ ❞❛❞♦ε >0✱ ❡♥❝♦♥✲

tr❛♠♦s δ >0 t❛❧ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r _kh_k< δ t❡♠♦s

kϕ′(u0 +h)−ϕ′(x0)k ≤ε.

P♦r ✭2.1✮✱

kϕ(u0+h)−ϕ(u0)−hϕ′(u0), hik ≤ sup

0≤θ_≤1k

ϕ′(u0+θ(h))−ϕ′(u0)kkhk ≤εkhk,

❞♦♥❞❡ s❡❣✉❡ q✉❡ ϕ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❛ ❋ré❝❤❡t ❡ ❡st❛ é ❝♦♥tí♥✉❛✳

✷✳✸ ❈♦♥str✉çã♦ ❞♦ ❈❛♠♣♦ Ps❡✉❞♦✲●r❛❞✐❡♥t❡

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♦ ❞❡♠♦♥str❛r♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ✉t✐✲ ❧✐③❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ❞❡✜♥✐❞♦ ♣♦r P❛❧❛✐s ❡♠ ✶✾✻✻✳

❉❡✜♥✐çã♦ ✹✳ ❙❡❥❛ M ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱X ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❡h :M →

X′_{\{0} ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❯♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ g é ✉♠ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡}

♣❛r❛ h ❡♠ M s❡ g : M _→ X é ❝♦♥tí♥✉♦ ❡ ❧♦❝❛❧♠❡♥t❡ ❞❡ ▲✐♣s❝❤✐t③ t❛❧ q✉❡

♣❛r❛ t♦❞♦ u∈M✱

kg(u)_{k ≤}2_kh(u)_k ✱ _hh(u), g(u)_{i ≥ k}h(u)_k2_.

◆♦t❡ q✉❡

kh(u)_k2 _{≤ h}h(u), g(u)_{i ≤ k}h(u)_kkg(u)_k

❡

kh(u)_{k ≤ k}g(u)_{k ≤}2_kh(u)_k,

♦♥❞❡

kh(u)_k= sup

kyk=1

y_∈X

hh(u), y_i.

▲❡♠❛ ✷✳✹✳ ✭❊①✐stê♥❝✐❛ ❞♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡✮ ❙❡❥❛ M ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ X

✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❡ h : M → X′\{0} ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ h ❡♠ M✳

❉❡♠♦♥str❛çã♦✿

❉❛❞♦ v _∈M✱ ❡①✐st❡✱ ♣♦r ❞❡✜♥✐çã♦✱x_∈M t❛❧ q✉❡ _kx_k= 1 ❡

hh(v), xi> 2

3kh(v)k. ✭✷✳✷✮

❉❡✜♥❛y:= 3

2kh(v)kx✳ ❆ss✐♠ t❡♠♦s

ky_k= 3

2kh(v)k<2kh(v)k

❡ hh(v), y_i= 3

2kh(v)khh(v), xi>kh(v)k

2_,

❡ ❡st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡♠♦s ❞❡ (2.2)✳ ❈♦♠♦ h é ❝♦♥tí♥✉❛✱ ❡①✐st❡ ✉♠❛

✈✐③✐♥❤❛♥ç❛ Nv ❞❡v t❛❧ q✉❡

kyk ≤2kh(u) ✱hh(u), yi ≥ kh(u)k2 _{♣❛r❛ t♦❞♦u∈N}

v. ✭✷✳✸✮

❆ ❢❛♠í❧✐❛ N _{:= {Nv;v ∈ M}} ♦❜✈✐❛♠❡♥t❡ é ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ M✳ ❈♦♠♦ M é ♠étr✐❝♦✱ ❧♦❣♦✱ ♣❛r❛❝♦♠♣❛❝t♦ ✭❱❡r ❊❧♦♥ ✲ ❊s♣❛ç♦s ▼étr✐❝♦s✱

♣❣✳ ✷✽✺✮✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ M _{= {Mi : i ∈ I}} ❞❡ _M✱ ❛❜❡rt❛ ❡

❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛✱ q✉❡ r❡✜♥❛ N✳ ■st♦ é✱ ♣❛r❛ ❝❛❞❛ _{u ∈ M} ❡①✐st❡♠ í♥❞✐❝❡s

λ1, . . . , λn ∈ I ❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ Wu ∋ u t❛✐s q✉❡ W ∩Mλ 6= ∅ ⇒ λ ∈

{λ1, . . . , λn}. ❊ ❛✐♥❞❛ ♣❛r❛ ❝❛❞❛ Mi ∈ M ❡①✐st❡ Nv ∈ N t❛❧ q✉❡ Mi ⊂ Nv✳

❆ss✐♠✱ ♣❛r❛ ❝❛❞❛ i✱ ♣♦❞❡♠♦s ❡❧❡❣❡r yi := y t❛❧ q✉❡ ✭✷✳✸✮ é s❛t✐s❢❡✐t♦ ♣❛r❛

❝❛❞❛ u∈Mi.

❙❡❥❛ ρi(u) ❛ ❞✐stâ♥❝✐❛ ❞❡ u ❛♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ Mi✳ ❊♥tã♦ ρi(u) é ❧✐♣s✲

❝❤✐t③✐❛♥❛ ❡ ρi(u)❂✵ s❡ u /∈Mi✳ ❉❡✜♥❛

g(u) :=X

i∈I

ρi(u)

P

j_∈Iρj(u)

yi.

❖❜s❡r✈❛♠♦s q✉❡ g ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s ♣❛r❛ ❝❛❞❛ u ∈ M ❛ s♦♠❛ ♥♦

❞❡♥♦♠✐♥❛❞♦r é ✜♥✐t❛ ✭✈✐st♦ q✉❡ ♦ r❡✜♥❛♠❡♥t♦ {Mi} é ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✮ ❡

❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳

❱❡r✐✜q✉❡♠♦s q✉❡g é ✉♠ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ h ❡♠ M✳

(i)▲♦❝❛❧♠❡♥t❡ ❞❡ ▲✐♣s❝❤✐t③✿ ✜①❛❞♦ u∈M✱ t♦♠❡♠♦sWu ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡

u q✉❡ ✐♥t❡r❝❡♣t❛ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❛❜❡rt♦s Mλ1, . . . , Mλn ❞❡ M✳ ❉❛❞♦s a ❡ b ❡♠ Wu✱

|g(a)₋g(b)_{| ≤}X

i_∈I

ρi(a)

P

j_∈Iρj(a)−

ρi(b)

P

j_∈Iρj(b)

|yi|

≤ X

i∈I

ρi(a)

P

j_∈Iρj(a)−

ρi(b)

P

j_∈Iρj(a)

+

ρi(b)

P

j_∈Iρj(a) −

ρi(b)

P

j_∈Iρj(b)

!

|yi|

≤ X

i∈I

"

k1|ρi(a)−ρi(b)|+|

ρi(b)|

k2

X

j∈I

|ρj(b)−ρj(a)|

#

|yi|,

♦♥❞❡

k1 = P 1

j_∈Iρj(a)

, k2 = _P 1

j_∈Iρj(a) Pj_∈Iρj(b)

.

▲♦❣♦✱ g é ▲✐♣s❝❤✐t③✳

(ii) g s❛t✐s❢❛③ ✭✷✳✸✮✿

kg(u)k ≤X

i∈I

ρi(u)

P

j_∈Iρj(u)

2kh(u)k ≤2kh(u)k,

hh(u), g(u)_{i ≥}X

i_∈I

ρi(u)

P

j∈Iρj(u)

kh(u)_k2 =_kh(u)_k2.

▲♦❣♦✱ g é ✉♠ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ h ❡♠ M✳

✷✳✺ ▲❡♠❛ ❞❡ ❞❡❢♦r♠❛çã♦ ❞❡ ❈❧❛r❦

❊st❛ ✈❡rsã♦ ❞♦ ▲❡♠❛ ❞❡ ❞❡❢♦r♠❛çã♦ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❲✐❧❧❡♠ ▼✳ ❡♠ ✶✾✽✸✱ ❞✉r❛♥t❡ s✉❛ ✈✐s✐t❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ [✶✾]✳

P❛r❛S _⊂X ❡ α >0 ❞❡✜♥❛ Sα :={u∈X;dist(u, S)≤α}✳

P❛r❛ϕ _∈C(X,R)✱ ❡d _∈R✱ ❞❡✜♥❛ ϕd _{=:{u∈X;ϕ(u)≤d}✳}

▲❡♠❛ ✷✳✻✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ ϕ∈C1_{(X,R)✱ S ⊂X✱ c∈R✱ ❡} ε >0, δ >0 t❛✐s q✉❡ ♣❛r❛ t♦❞♦

u_∈ϕ−1[c₋2ε, c+ 2ε]_∩S2δ, t❡♠♦s kϕ′(u)k ≥

8ε δ .

❊♥tã♦ ❡①✐st❡ η∈C1_{([0,1]×X, X) t❛❧ q✉❡}

✭✐✮ η(t, u) =u s❡ t= 0 ♦✉ s❡ u /_∈ϕ−1_{([c−2ε, c+ 2ε])∩S2}

δ✱

✭✐✐✮ η(t, ϕc+ε_∩S)⊂ϕc₋ε_✱

✭✐✐✐✮ η(t,_·) :X _→X é ❤♦♠❡♦♠♦r✜s♠♦ ❞❡ X ❡♠ X ♣❛r❛ t♦s♦ t_∈[0,1]✱

✭✐✈✮ kη(t, u)−uk ≤δ✱ ∀ u∈X✱ ∀ t ∈[0,1]✱

✭✈✮ ϕ(η(_·, u))é ♥ã♦ ❝r❡s❝❡♥t❡ ♣❛r❛ t♦❞♦ u_∈X✱

✭✈✐✮ ϕ(η(t, u))< c✱ ♣❛r❛ t♦❞♦ u∈ϕc_∩S

δ✱ ❡ t♦❞♦ t∈(0,1]✳

❉❡♠♦♥str❛çã♦✿

P❡❧♦ ▲❡♠❛ 2.4✱ ❡①✐st❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡g ♣❛r❛ ϕ′ _❡♠

M :={u∈X;ϕ′(u)6= 0}

❝♦♠ kϕ′(u)_{k ≤ k}g(u)_{k ≤}2_kϕ′(u)_k. ✭✷✳✹✮

❱❛♠♦s ❞❡✜♥✐r

A:=ϕ−1([c₋2ε, c+ 2ε])_∩S2δ,

B :=ϕ−1([c−ε, c+ε])∩Sδ.

❈❧❛r❛♠❡♥t❡ A ❡ B sã♦ ❢❡❝❤❛❞♦s ❡ B A✳ ❆ss✐♠✱ dist(u, X_\A) ❡ dist(u, B)

♥ã♦ sã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥✉❧♦s✱ ❡ ♠❛✐s✱ ❞❛❞♦ u ∈ X, ❡①✐st❡♠ c✱ d ❡♠ R ❡

✉♠❛ ✈✐③✐♥❤❛♥ç❛ Wu ∋u t❛✐s q✉❡ s❡ v ∈Wu

0< c _≤dist(v, X_\A) +dist(v, B)_≤d <_∞.

❉❡✜♥❛Ψ :X→R ♣♦♥❞♦

Ψ(u) := dist(u, X\A)

dist(u, X_\A) +dist(u, B).

◆♦t❡ q✉❡ Ψ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ✭♣♦✐s ♦ ❞❡♥♦♠✐♥❛❞♦r é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ♣❛r❛

t♦❞♦ u_∈X✮ ❡ q✉❡ 0_≤Ψ_≤1 ❝♦♠

Ψ(u) = 0 s❡u_∈B,

Ψ(u) = 1 s❡u∈X\A.

❈♦♠♦ dist(u, X_\A) ❡ dist(u, B) sã♦ ❧✐♣s❝❤✐t③✱ ❛♥á❧♦❣♦ à ❞❡♠♦♥str❛çã♦ ❞♦

♣s❡✉❞♦✲❣r❛❞✐❡♥t❡✱ t❡♠♦s q✉❡ Ψé ▲✐♣s❝❤✐t③ ❝♦♥tí♥✉❛✳

❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❡♠X ❞❡✜♥✐❞♦ ♣♦r

f(u) :=

−Ψ(u)_kg(u)_k−2_{g(u) , u∈A}

0 u∈X\A ✭✷✳✺✮

◆♦t❡ q✉❡ A_⊂M ✭❧♦❣♦ ♣♦❞❡♠♦s ❞❡✜♥✐rg ❡♠ A✮✱ ❡ ♣♦r ✭2.4✮ t❡♠♦s q✉❡

kf(u)k ≤ 1 kg(u)_k ≤

1

kϕ′_(u)k ≤

δ

8ε, ♣❛r❛ t♦❞♦ u∈X.

❊ ♣♦rt❛♥t♦✱ f ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳

❱❡r✐✜q✉❡♠♦s q✉❡ f é ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛✳ ❉❛❞♦ u ∈ X✱ t♦♠❡♠♦s Wu ✈✐③✐♥❤❛♥ç❛ ❞❡ ut❛❧ q✉❡ g|Wu ❡ Ψ|Wu sã♦ ❧✐♣s❝❤✐t③✳ ❉❛❞♦sa✱ b ❡♠ Wu✳

•❙❡ a✱ b ∈X\A✱ t❡♠♦s

kf(a)₋f(b)_k= 0 _{≤ k}a₋b_k.