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r rt r t r - - PowerPoint PPT Presentation

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r rt r t r r t s Pr tr r

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SLIDE 1

❇♦✉♥❞❛r② ❘❡❣✉❧❛r✐t② ❢♦r t❤❡ ❋r❡❡ ❇♦✉♥❞❛r② ✐♥ t❤❡ ❖♥❡✲♣❤❛s❡ Pr♦❜❧❡♠

❍é❝t♦r ❆✳ ❈❤❛♥❣✲▲❛r❛ ✭❈■▼❆❚ ✲ ●✉❛♥❛❥❛✉t♦✱ ▼❡①✐❝♦✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❖✈✐❞✐✉ ❙❛✈✐♥ ✭❈♦❧✉♠❜✐❛ ❯♥✐✈❡rs✐t②✮ ❙✇❡❞✐s❤ ❙✉♠♠❡r P❉❊s ❆✉❣✉st ✷✻✲✷✽✱ ❑❚❍

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SLIDE 2

❆ t❛❧❦ ❛❜♦✉t ❢♦✉♥t❛✐♥s

  • ♦❛❧✿ ❉❡s❝r✐❜❡ ❤♦✇ t❤❡ ❥❡t s❡♣❛r❛t❡s✳

✶ ✴ ✷✵

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SLIDE 3

❆ t❛❧❦ ❛❜♦✉t ❢♦✉♥t❛✐♥s

  • ♦❛❧✿ ❉❡s❝r✐❜❡ ❤♦✇ t❤❡ ❥❡t s❡♣❛r❛t❡s✳

✶ ✴ ✷✵

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SLIDE 4

❏❡ts

■♥❝♦♠♣r❡ss✐❜✐❧✐t②✿ ❊①✐sts str❡❛♠ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ✈❡❧♦❝✐t② ■rr♦t❛t✐♦♥❛❧✐t②✿ ❇❡r♥♦✉❧❧✐✬s ❧❛✇✿ ✭ ♦r ✮ ♦✈❡r

✷ ✴ ✷✵

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SLIDE 5

❏❡ts

■♥❝♦♠♣r❡ss✐❜✐❧✐t②✿ ❊①✐sts str❡❛♠ ❢✉♥❝t✐♦♥ u ≥ 0 s✉❝❤ t❤❛t ✈❡❧♦❝✐t② = Du⊥ = (∂2u, −∂1u) ■rr♦t❛t✐♦♥❛❧✐t②✿ ❇❡r♥♦✉❧❧✐✬s ❧❛✇✿ ✭ ♦r ✮ ♦✈❡r

✷ ✴ ✷✵

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SLIDE 6

❏❡ts

■♥❝♦♠♣r❡ss✐❜✐❧✐t②✿ ❊①✐sts str❡❛♠ ❢✉♥❝t✐♦♥ u ≥ 0 s✉❝❤ t❤❛t ✈❡❧♦❝✐t② = Du⊥ = (∂2u, −∂1u) ■rr♦t❛t✐♦♥❛❧✐t②✿ ∆u = 0 ❇❡r♥♦✉❧❧✐✬s ❧❛✇✿ ✭ ♦r ✮ ♦✈❡r

✷ ✴ ✷✵

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SLIDE 7

❏❡ts

■♥❝♦♠♣r❡ss✐❜✐❧✐t②✿ ❊①✐sts str❡❛♠ ❢✉♥❝t✐♦♥ u ≥ 0 s✉❝❤ t❤❛t ✈❡❧♦❝✐t② = Du⊥ = (∂2u, −∂1u) ■rr♦t❛t✐♦♥❛❧✐t②✿ ∆u = 0 ❇❡r♥♦✉❧❧✐✬s ❧❛✇✿ |Du| = 1 ✭Du = −ν ♦r ∂νu = −1✮ ♦✈❡r Γ+

✷ ✴ ✷✵

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SLIDE 8

❖♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❆❧t✲❈❛✛❛r❡❧❧✐ ✶✾✽✶✿ ■♥t❡r✐♦r r❡s✉❧ts ❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ✐s r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s

✸ ✴ ✷✵

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SLIDE 9

❖♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❆❧t✲❈❛✛❛r❡❧❧✐ ✶✾✽✶✿ ■♥t❡r✐♦r r❡s✉❧ts ❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ✐s r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s

✸ ✴ ✷✵

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SLIDE 10

❖♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❆❧t✲❈❛✛❛r❡❧❧✐ ✶✾✽✶✿ ■♥t❡r✐♦r r❡s✉❧ts ◮ u ∈ C0,1

loc

❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ✐s r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s

✸ ✴ ✷✵

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SLIDE 11

❖♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❆❧t✲❈❛✛❛r❡❧❧✐ ✶✾✽✶✿ ■♥t❡r✐♦r r❡s✉❧ts ◮ u ∈ C0,1

loc

◮ Ω+ ❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ✐s r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s

✸ ✴ ✷✵

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SLIDE 12

❖♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❆❧t✲❈❛✛❛r❡❧❧✐ ✶✾✽✶✿ ■♥t❡r✐♦r r❡s✉❧ts ◮ u ∈ C0,1

loc

◮ Ω+ ❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ◮ Γ+ ✐s C1,α r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s

✸ ✴ ✷✵

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SLIDE 13

❏❡ts ❛♥❞ ❝❛✈✐t✐❡s

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❙❧✐♣ ❝♦♥❞✐t✐♦♥✿ ♦✈❡r ❍♦✇ ❞♦❡s ❞❡t❛❝❤ ❢r♦♠ ❄

✹ ✴ ✷✵

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SLIDE 14

❏❡ts ❛♥❞ ❝❛✈✐t✐❡s

u ≥ 0 s❛t✐s✜❡s

  • ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω ❙❧✐♣ ❝♦♥❞✐t✐♦♥✿ u = 0 ♦✈❡r Z = {x2 > 0} ❍♦✇ ❞♦❡s Γ ❞❡t❛❝❤ ❢r♦♠ Z❄

✹ ✴ ✷✵

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SLIDE 15

❘❡❣✉❧❛r✐t② ♦❢ Γ ✉♣ t♦ t❤❡ ✭✜①❡❞✮ ❜♦✉♥❞❛r②

❚❤❡♦r❡♠ ✭❈✲❙❛✈✐♥✮

▲❡t Ω ⊆ Rn ❜❡ ❛ ❞♦♠❛✐♥ ✇✐t❤ ❛ C1,α ❜♦✉♥❞❛r② ♣♦rt✐♦♥ Z ⊆ ∂Ω ✇✐t❤ α > 1/2✳ ▲❡t u ≥ 0 ❜❡ ❛ ✭✈✐s❝♦s✐t②✮ s♦❧✉t✐♦♥ ♦❢            ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω |Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω u = 0 ♦♥ Z |Du| ≥ 1 ♦♥ Γ0 = ∂Ω+ ∩ Z ❚❤❡♥ Γ = Γ+ ∪ Γ0 ✐s C1,1/2 r❡❣✉❧❛r ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❡✈❡r② x0 ∈ Γ0✳

✺ ✴ ✷✵

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SLIDE 16

❘❡❣✉❧❛r✐t② ♦❢ Γ ✉♣ t♦ t❤❡ ✭✜①❡❞✮ ❜♦✉♥❞❛r②

❚❤❡♦r❡♠ ✭❈✲❙❛✈✐♥✮

▲❡t Ω ⊆ Rn ❜❡ ❛ ❞♦♠❛✐♥ ✇✐t❤ ❛ C1,α ❜♦✉♥❞❛r② ♣♦rt✐♦♥ Z ⊆ ∂Ω ✇✐t❤ α > 1/2✳ ▲❡t u ≥ 0 ❜❡ ❛ ✭✈✐s❝♦s✐t②✮ s♦❧✉t✐♦♥ ♦❢            ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ Ω |Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ Ω u = 0 ♦♥ Z |Du| ≥ 1 ♦♥ Γ0 = ∂Ω+ ∩ Z ❚❤❡♥ Γ = Γ+ ∪ Γ0 ✐s C1,1/2 r❡❣✉❧❛r ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❡✈❡r② x0 ∈ Γ0✳

✺ ✴ ✷✵

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SLIDE 17

✭ ¯ F = Γ✮

✻ ✴ ✷✵

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SLIDE 18

|Du| ≥ 1 ♦♥ Γ0❄

❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ ✐♥ ❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ ❚❤✐s ✐♠♣❧✐❡s ♦♥ ❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ ♦♥

✼ ✴ ✷✵

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SLIDE 19

|Du| ≥ 1 ♦♥ Γ0❄

❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ min{Ju : u = g ≥ 0 ✐♥ ∂Ω} Ju = ˆ

Ω

|Du|2 + χ{u>0} ❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ ❚❤✐s ✐♠♣❧✐❡s ♦♥ ❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ ♦♥

✼ ✴ ✷✵

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SLIDE 20

|Du| ≥ 1 ♦♥ Γ0❄

❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ min{Ju : u = g ≥ 0 ✐♥ ∂Ω} Ju = ˆ

Ω

|Du|2 + χ{u>0} ❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ uε(x + εη(x)) = u(x) Juε = Ju + ε ˆ

Γ

(1 − |Du|2)η · ν + o(ε) ❚❤✐s ✐♠♣❧✐❡s |Du| = 1 ♦♥ Γ+ ❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ ♦♥

✼ ✴ ✷✵

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SLIDE 21

|Du| ≥ 1 ♦♥ Γ0❄

❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ min{Ju : u = g ≥ 0 ✐♥ ∂Ω} Ju = ˆ

Ω

|Du|2 + χ{u>0} ❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ uε(x + εη(x)) = u(x) Juε = Ju + ε ˆ

Γ

(1 − |Du|2)η · ν + o(ε) ❚❤✐s ✐♠♣❧✐❡s |Du| = 1 ♦♥ Γ+ ❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ Z |Du| ≥ 1 ♦♥ Γ0

✼ ✴ ✷✵

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SLIDE 22

|Du| ≥ 1 ♦♥ Γ0❄

❙♦❧✉t✐♦♥s ❝❛♥ ❛❧s♦ ❜❡ ❝♦♥tr❛st❡❞ ❜② P❡rr♦♥✬s ♠❡t❤♦❞✿ u ✐s t❤❡ s♠❛❧❧❡st s✉♣❡rs♦❧✉t✐♦♥ ❛❜♦✈❡ ❛ s✉❜s♦❧✉t✐♦♥ t❛❦✐♥❣ t❤❡ ❜♦✉♥❞❛r② ❞❛t✉♠ g ≥ 0✳ ❆❧s♦ ✐♥ t❤✐s s❡tt✐♥❣ ✇❡ r❡❝♦✈❡r ♦♥

✽ ✴ ✷✵

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SLIDE 23

|Du| ≥ 1 ♦♥ Γ0❄

❙♦❧✉t✐♦♥s ❝❛♥ ❛❧s♦ ❜❡ ❝♦♥tr❛st❡❞ ❜② P❡rr♦♥✬s ♠❡t❤♦❞✿ u ✐s t❤❡ s♠❛❧❧❡st s✉♣❡rs♦❧✉t✐♦♥ ❛❜♦✈❡ ❛ s✉❜s♦❧✉t✐♦♥ t❛❦✐♥❣ t❤❡ ❜♦✉♥❞❛r② ❞❛t✉♠ g ≥ 0✳ ❆❧s♦ ✐♥ t❤✐s s❡tt✐♥❣ ✇❡ r❡❝♦✈❡r |Du| ≥ 1 ♦♥ Γ0

✽ ✴ ✷✵

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SLIDE 24

Γ ∈ C1,1/2❄

▲❡t Ω = B+

1 ❛♥❞ Z = B′ 1✳

❆ss✉♠✐♥❣ t❤❛t ❞❡t❛❝❤❡s ❢r♦♠ ❛t ✇❡ ❧❡t ✳ ❈❧❡❛r❧② ✐s ❤❛r♠♦♥✐❝ ✐♥ ✱ ❛♥❞ ♥♦♥♥❡❣❛t✐✈❡ ♦✈❡r ✳ ❍♦✇ ❞♦❡s t❤❡ ❢r❡❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ r❡✢❡❝t ♦♥ ❄ ♠❡❛♥s t❤❛t ♦♥ ❆❞❞✐t✐♦♥❛❧❧②✱ ♦♥ s❛②s t❤❛t ♦♥

✾ ✴ ✷✵

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SLIDE 25

Γ ∈ C1,1/2❄

▲❡t Ω = B+

1 ❛♥❞ Z = B′ 1✳

❆ss✉♠✐♥❣ t❤❛t Γ ❞❡t❛❝❤❡s ❢r♦♠ Z ❛t 0 ✇❡ ❧❡t u = xn − εw ✳ ❈❧❡❛r❧② w ✐s ❤❛r♠♦♥✐❝ ✐♥ Ω+✱ ❛♥❞ ♥♦♥♥❡❣❛t✐✈❡ ♦✈❡r Γ✳ ❍♦✇ ❞♦❡s t❤❡ ❢r❡❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ r❡✢❡❝t ♦♥ w❄ ♠❡❛♥s t❤❛t ♦♥ ❆❞❞✐t✐♦♥❛❧❧②✱ ♦♥ s❛②s t❤❛t ♦♥

✾ ✴ ✷✵

slide-26
SLIDE 26

Γ ∈ C1,1/2❄

▲❡t Ω = B+

1 ❛♥❞ Z = B′ 1✳

❆ss✉♠✐♥❣ t❤❛t Γ ❞❡t❛❝❤❡s ❢r♦♠ Z ❛t 0 ✇❡ ❧❡t u = xn − εw ✳ ❈❧❡❛r❧② w ✐s ❤❛r♠♦♥✐❝ ✐♥ Ω+✱ ❛♥❞ ♥♦♥♥❡❣❛t✐✈❡ ♦✈❡r Γ✳ ❍♦✇ ❞♦❡s t❤❡ ❢r❡❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ r❡✢❡❝t ♦♥ w❄ Du = −ν ♠❡❛♥s t❤❛t ∂nw = 1 ε(1 + νn) = ε 2|Dw|2 ♦♥ Γ+. ❆❞❞✐t✐♦♥❛❧❧②✱ |Du| ≥ 1 ♦♥ Γ0 s❛②s t❤❛t ∂nw ≥ 0 ♦♥ Γ0.

✾ ✴ ✷✵

slide-27
SLIDE 27

Γ ∈ C1,1/2❄

❆s ε → 0✱ ✇❡ ❡①♣❡❝t Ω+ → B+

1 ✱ ❛♥❞ w t♦ s♦❧✈❡

           ∆w = 0 ✐♥ B+

1

w ≥ 0 ♦♥ B′

1

∂nw = 0 ♦♥ {w > 0} ∩ B′

1

∂nw ≤ 0 ♦♥ B′

1

❚❤❡ ❙✐❣♥♦r✐♥✐ ♦r t❤✐♥ ♦❜st❛❝❧❡ ♣r♦❜❧❡♠✦

❋✐❣✉r❡✿ ■♠❛❣❡ ❜② ❆rs❤❛❦ P❡tr♦s②❛♥

✶✵ ✴ ✷✵

slide-28
SLIDE 28

Γ ∈ C1,1/2❄

❆s ε → 0✱ ✇❡ ❡①♣❡❝t Ω+ → B+

1 ✱ ❛♥❞ w t♦ s♦❧✈❡

           ∆w = 0 ✐♥ B+

1

w ≥ 0 ♦♥ B′

1

∂nw = 0 ♦♥ {w > 0} ∩ B′

1

∂nw ≤ 0 ♦♥ B′

1

❚❤❡ s❛♠❡ ❧✐♥❡❛r✐③❛t✐♦♥ ✐s ❢♦✉♥❞ ❛♥❞ ✉s❡❞ ❜② ❆♥❞❡rss♦♥✱ ❙❤❛❤❣❤♦❧✐❛♥✱ ❛♥❞ ❲❡✐ss✳ ❆t❤❛♥❛s♦♣♦✉❧♦s✲❈❛✛❛r❡❧❧✐ ✷✵✵✹✿ ✐s t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✳ ■♥❞❡❡❞ ✐s ❛ s♦❧✉t✐♦♥✳ ❖✉r r❡s✉❧t s❛②s t❤❛t ✐♥❤❡r✐ts t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❙✐❣♥♦r✐♥✐ ♣r♦❜❧❡♠✳

✶✶ ✴ ✷✵

slide-29
SLIDE 29

Γ ∈ C1,1/2❄

❆s ε → 0✱ ✇❡ ❡①♣❡❝t Ω+ → B+

1 ✱ ❛♥❞ w t♦ s♦❧✈❡

           ∆w = 0 ✐♥ B+

1

w ≥ 0 ♦♥ B′

1

∂nw = 0 ♦♥ {w > 0} ∩ B′

1

∂nw ≤ 0 ♦♥ B′

1

❚❤❡ s❛♠❡ ❧✐♥❡❛r✐③❛t✐♦♥ ✐s ❢♦✉♥❞ ❛♥❞ ✉s❡❞ ❜② ❆♥❞❡rss♦♥✱ ❙❤❛❤❣❤♦❧✐❛♥✱ ❛♥❞ ❲❡✐ss✳ ❆t❤❛♥❛s♦♣♦✉❧♦s✲❈❛✛❛r❡❧❧✐ ✷✵✵✹✿ w ∈ C1,1/2 ✐s t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✳ ■♥❞❡❡❞ w = r3/2 cos(3θ/2) ✐s ❛ s♦❧✉t✐♦♥✳ ❖✉r r❡s✉❧t s❛②s t❤❛t ✐♥❤❡r✐ts t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❙✐❣♥♦r✐♥✐ ♣r♦❜❧❡♠✳

✶✶ ✴ ✷✵

slide-30
SLIDE 30

Γ ∈ C1,1/2❄

❆s ε → 0✱ ✇❡ ❡①♣❡❝t Ω+ → B+

1 ✱ ❛♥❞ w t♦ s♦❧✈❡

           ∆w = 0 ✐♥ B+

1

w ≥ 0 ♦♥ B′

1

∂nw = 0 ♦♥ {w > 0} ∩ B′

1

∂nw ≤ 0 ♦♥ B′

1

❚❤❡ s❛♠❡ ❧✐♥❡❛r✐③❛t✐♦♥ ✐s ❢♦✉♥❞ ❛♥❞ ✉s❡❞ ❜② ❆♥❞❡rss♦♥✱ ❙❤❛❤❣❤♦❧✐❛♥✱ ❛♥❞ ❲❡✐ss✳ ❆t❤❛♥❛s♦♣♦✉❧♦s✲❈❛✛❛r❡❧❧✐ ✷✵✵✹✿ w ∈ C1,1/2 ✐s t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✳ ■♥❞❡❡❞ w = r3/2 cos(3θ/2) ✐s ❛ s♦❧✉t✐♦♥✳ ❖✉r r❡s✉❧t s❛②s t❤❛t Γ = {xn = εw} ✐♥❤❡r✐ts t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❙✐❣♥♦r✐♥✐ ♣r♦❜❧❡♠✳

✶✶ ✴ ✷✵

slide-31
SLIDE 31

❙tr❛t❡❣②

❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②✿ ❈♦♠♣❛❝t♥❡ss s♦ ✇❡ ❝❛♥ ❧✐♥❡❛r✐s❡✳ ❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❋♦r ✱ ❜② ❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥t✳ ❖♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❜② ❛♥ ❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛✳ ❋r♦♠ ♥♦✇ ♦♥ ✱ ❛♥❞ s❛t✐s✜❡s ✐♥ ♦♥ ♦♥ ♦♥

✶✷ ✴ ✷✵

slide-32
SLIDE 32

❙tr❛t❡❣②

◮ ❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②✿ ❈♦♠♣❛❝t♥❡ss s♦ ✇❡ ❝❛♥ ❧✐♥❡❛r✐s❡✳ ❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❋♦r ✱ ❜② ❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥t✳ ❖♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❜② ❛♥ ❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛✳ ❋r♦♠ ♥♦✇ ♦♥ ✱ ❛♥❞ s❛t✐s✜❡s ✐♥ ♦♥ ♦♥ ♦♥

✶✷ ✴ ✷✵

slide-33
SLIDE 33

❙tr❛t❡❣②

◮ ❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②✿ ❈♦♠♣❛❝t♥❡ss s♦ ✇❡ ❝❛♥ ❧✐♥❡❛r✐s❡✳ ◮ ❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❋♦r β ∈ (0, 1/2)✱ Γ ∈ C1,β ❜② ❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥t✳ ❖♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❜② ❛♥ ❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛✳ ❋r♦♠ ♥♦✇ ♦♥ ✱ ❛♥❞ s❛t✐s✜❡s ✐♥ ♦♥ ♦♥ ♦♥

✶✷ ✴ ✷✵

slide-34
SLIDE 34

❙tr❛t❡❣②

◮ ❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②✿ ❈♦♠♣❛❝t♥❡ss s♦ ✇❡ ❝❛♥ ❧✐♥❡❛r✐s❡✳ ◮ ❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❋♦r β ∈ (0, 1/2)✱ Γ ∈ C1,β ❜② ❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥t✳ ◮ ❖♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ Γ ∈ C1,1/2 ❜② ❛♥ ❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛✳ ❋r♦♠ ♥♦✇ ♦♥ ✱ ❛♥❞ s❛t✐s✜❡s ✐♥ ♦♥ ♦♥ ♦♥

✶✷ ✴ ✷✵

slide-35
SLIDE 35

❙tr❛t❡❣②

◮ ❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②✿ ❈♦♠♣❛❝t♥❡ss s♦ ✇❡ ❝❛♥ ❧✐♥❡❛r✐s❡✳ ◮ ❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ ❋♦r β ∈ (0, 1/2)✱ Γ ∈ C1,β ❜② ❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥t✳ ◮ ❖♣t✐♠❛❧ r❡❣✉❧❛r✐t②✿ Γ ∈ C1,1/2 ❜② ❛♥ ❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛✳ ❋r♦♠ ♥♦✇ ♦♥ Ω = B+

1 ✱ Z = B′ 1 ❛♥❞ u s❛t✐s✜❡s

           ∆u = 0 ✐♥ Ω+ = {u > 0} ∩ B+

1

|Du| = 1 ♦♥ Γ+ = ∂{u > 0} ∩ B+

1

u = 0 ♦♥ B′

1

|Du| ≥ 1 ♦♥ Γ0 = ∂Ω+ ∩ B′

1

✶✷ ✴ ✷✵

slide-36
SLIDE 36

Pr❡❧✐♠✐♥❛r✐❡s✿ ❋r❡❡ ✢❛t♥❡ss

▲✐♥❡❛r ❜❡❤❛✈✐♦r ❛t r❡❣✉❧❛r ❜♦✉♥❞❛r② ♣♦✐♥ts✿ v ∈ C(Br)✱ ♥♦♥♥❡❣❛t✐✈❡✱ ❤❛r♠♦♥✐❝ ♦✈❡r Ω+ = {v > 0}✱ ❛♥❞ 0 ∈ ∂Ω+✳ ❆♥ ❡①t❡r✐♦r ❜❛❧❧ ❝♦♥❞✐t✐♦♥ ❢♦r Ω+ ❛t 0 ✐♠♣❧✐❡s v(x) = α(x · ν)+ + o(|x|) ❛s x ∈ Ω+ → 0 ♥♦♥t❛♥❣❡♥t✐❛❧❧② ■❢ ❞❡t❛❝❤❡s ❢r♦♠ ❛t ✱ t❤❡♥ ❛❢t❡r ❛ r❡s❝❛❧✐♥❣ ✇❡ ❝❛♥ ❛ss✉♠❡ ✐♥ ❢♦r ❛s s♠❛❧❧ ❛s ✇❡ ✇✐s❤✳

✶✸ ✴ ✷✵

slide-37
SLIDE 37

Pr❡❧✐♠✐♥❛r✐❡s✿ ❋r❡❡ ✢❛t♥❡ss

▲✐♥❡❛r ❜❡❤❛✈✐♦r ❛t r❡❣✉❧❛r ❜♦✉♥❞❛r② ♣♦✐♥ts✿ v ∈ C(Br)✱ ♥♦♥♥❡❣❛t✐✈❡✱ ❤❛r♠♦♥✐❝ ♦✈❡r Ω+ = {v > 0}✱ ❛♥❞ 0 ∈ ∂Ω+✳ ❆♥ ❡①t❡r✐♦r ❜❛❧❧ ❝♦♥❞✐t✐♦♥ ❢♦r Ω+ ❛t 0 ✐♠♣❧✐❡s v(x) = α(x · ν)+ + o(|x|) ❛s x ∈ Ω+ → 0 ♥♦♥t❛♥❣❡♥t✐❛❧❧② ■❢ Γ ❞❡t❛❝❤❡s ❢r♦♠ Z ❛t 0✱ t❤❡♥ ❛❢t❡r ❛ r❡s❝❛❧✐♥❣ ✇❡ ❝❛♥ ❛ss✉♠❡ (1 + ε)xn ≥ u ≥ xn − ε ✐♥ B+

1

❢♦r ε > 0 ❛s s♠❛❧❧ ❛s ✇❡ ✇✐s❤✳

✶✸ ✴ ✷✵

slide-38
SLIDE 38

❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②

▲❡♠♠❛

❚❤❡r❡ ❡①✐st ε0, θ ∈ (0, 1) s✉❝❤ t❤❛t ✐❢ ❢♦r ε ∈ (0, ε0) (1 + ε)xn ≥ u ≥ xn − ε ✐♥ B+

1

❚❤❡♥ ✐♥ B+

1/2 ❡✐t❤❡r

xn ≥ u ♦r u ≥ xn − (1 − θ)ε

✶✹ ✴ ✷✵

slide-39
SLIDE 39

❍❛r♥❛❝❦✲t②♣❡ ✐♥❡q✉❛❧✐t②

Pr♦♦❢ ❜❛s❡❞ ♦♥

▲❡♠♠❛ ✭❉❡ ❙✐❧✈❛ ✷✵✶✶✮

▲❡t v ❜❡ ❛ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ t❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ✐♥ B1✳ ❚❤❡r❡ ❡①✐st ε0, θ ∈ (0, 1) s✉❝❤ t❤❛t ✐❢ ❢♦r a, b ∈ (0, ε0)✱ xn + a ≥ v ≥ xn − b ✐♥ B1 t❤❡♥ ✐♥ B1/2 ❡✐t❤❡r xn + a − θc ≥ v ♦r v ≥ xn − b + θc (c = (a + b)/2)

✶✺ ✴ ✷✵

slide-40
SLIDE 40

❈♦♠♣❛❝t♥❡ss

❈♦♥s✐❞❡r ❢♦r εk → 0 ❛ s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s {uk} s❛t✐s❢②✐♥❣ (1 + εk)xn ≥ u ≥ xn − εk ✐♥ B+

1

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t wk = xn − uk εk , Gk = {(x, y) ∈ Dk × R : y = wk(x)} ✇❤❡r❡ Dk = (Ω+

k ∪ Fk) ∩ B1/2✳

❈♦r♦❧❧❛r②

❚❤❡r❡ ❡①✐sts w ∈ C(B+

1/2 ∪ B′ 1/2) ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙✐❣♥♦r✐♥✐

♣r♦❜❧❡♠ s✉❝❤ t❤❛t ❛ s✉❜s❡q✉❡♥❝❡ ❢r♦♠ {Gk} ❝♦♥✈❡r❣❡s t♦ {(x, y) ∈ (B+

1/2 ∪ B′ 1/2) × R : y = w(x)} ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

❍❛✉s❞♦r✛ ❞✐st❛♥❝❡✳

✶✻ ✴ ✷✵

slide-41
SLIDE 41

❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②

▲❡♠♠❛

  • ✐✈❡♥ β ∈ (0, 1/2)✱ t❤❡r❡ ❡①✐st ε0, µ ∈ (0, 1) s✉❝❤ t❤❛t ✐❢ 0 ∈ Λ

❛♥❞ ❢♦r ε ∈ (0, ε0) (1 + ε)xn ≥ u ≥ xn − ε ✐♥ B+

1

t❤❡♥ u ≥ xn − εµ1+β ✐♥ B+

µ

❚❤✐s ✐♠♣❧✐❡s ❛ ♠♦❞✉❧✉s ♦❢ ❝♦♥t✐♥✉✐t② ❢♦r ❛t

✶✼ ✴ ✷✵

slide-42
SLIDE 42

❆❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t②

▲❡♠♠❛

  • ✐✈❡♥ β ∈ (0, 1/2)✱ t❤❡r❡ ❡①✐st ε0, µ ∈ (0, 1) s✉❝❤ t❤❛t ✐❢ 0 ∈ Λ

❛♥❞ ❢♦r ε ∈ (0, ε0) (1 + ε)xn ≥ u ≥ xn − ε ✐♥ B+

1

t❤❡♥ u ≥ xn − εµ1+β ✐♥ B+

µ

❚❤✐s ✐♠♣❧✐❡s ❛ C1,β ♠♦❞✉❧✉s ♦❢ ❝♦♥t✐♥✉✐t② ❢♦r Γ ❛t 0

✶✼ ✴ ✷✵

slide-43
SLIDE 43

❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛

❆ss✉♠❡ 0 ∈ Γ0 ❛♥❞ ❧❡t w = xn − u✳ ❖✉r ❣♦❛❧✱ s❤♦✇✐♥❣ t❤❛t Γ = {xn = w} ∈ C1,1/2(0) ✇✐❧❧ ❢♦❧❧♦✇ ❢r♦♠ H(r) :=

  • ∂Br∩Ω+ w2

1/2 ≤ Cr3/2 ■❢ ✇❡ ❝❛♥ ❝❛♣t✉r❡ t❤❡ ❢r❡q✉❡♥❝② ❢r♦♠ ❯s✐♥❣ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts

✶✽ ✴ ✷✵

slide-44
SLIDE 44

❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛

❆ss✉♠❡ 0 ∈ Γ0 ❛♥❞ ❧❡t w = xn − u✳ ❖✉r ❣♦❛❧✱ s❤♦✇✐♥❣ t❤❛t Γ = {xn = w} ∈ C1,1/2(0) ✇✐❧❧ ❢♦❧❧♦✇ ❢r♦♠ H(r) :=

  • ∂Br∩Ω+ w2

1/2 ≤ Cr3/2 ■❢ H(r) = Crk ✇❡ ❝❛♥ ❝❛♣t✉r❡ t❤❡ ❢r❡q✉❡♥❝② k ❢r♦♠ k = N(r) := r d dr ln H(r) = r d

dr

ffl

∂Br∩Ω+ w2

2 ffl

∂Br∩Ω+ w2

❯s✐♥❣ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts

✶✽ ✴ ✷✵

slide-45
SLIDE 45

❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛

❆ss✉♠❡ 0 ∈ Γ0 ❛♥❞ ❧❡t w = xn − u✳ ❖✉r ❣♦❛❧✱ s❤♦✇✐♥❣ t❤❛t Γ = {xn = w} ∈ C1,1/2(0) ✇✐❧❧ ❢♦❧❧♦✇ ❢r♦♠ H(r) :=

  • ∂Br∩Ω+ w2

1/2 ≤ Cr3/2 ■❢ H(r) = Crk ✇❡ ❝❛♥ ❝❛♣t✉r❡ t❤❡ ❢r❡q✉❡♥❝② k ❢r♦♠ k = N(r) := r d dr ln H(r) = r d

dr

ffl

∂Br∩Ω+ w2

2 ffl

∂Br∩Ω+ w2

❯s✐♥❣ ∆w = 0 ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts d dr

∂Br∩Ω+ w2 = ∂Br∩Ω+ 2wDw · ν + error

= 2r

Br∩Ω+ |Dw|2 + error

✶✽ ✴ ✷✵

slide-46
SLIDE 46

❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛

■♥ ❣❡♥❡r❛❧✱ ✐❢ H(r) ≤ Cr3/2 ✐t ♠❛❦❡s s❡♥s❡ t♦ ❡①♣❡❝t lim

r→0+ N(r) =

r d

dr

ffl

∂Br∩Ω+ w2

2 ffl

∂Br∩Ω+ w2

≥ 3/2

▲❡♠♠❛

❚❤❡r❡ ❡①✐st s✉❝❤ t❤❛t ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✇❤❡r❡ ❍❡r❡ t❤❡ ❛❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② ✐s ❝r✉❝✐❛❧ ✐♥ ♦r❞❡r t♦ ❥✉st✐❢② t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✶✾ ✴ ✷✵

slide-47
SLIDE 47

❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛

■♥ ❣❡♥❡r❛❧✱ ✐❢ H(r) ≤ Cr3/2 ✐t ♠❛❦❡s s❡♥s❡ t♦ ❡①♣❡❝t lim

r→0+ N(r) =

r d

dr

ffl

∂Br∩Ω+ w2

2 ffl

∂Br∩Ω+ w2

≥ 3/2

▲❡♠♠❛

❚❤❡r❡ ❡①✐st ε, η > 0 s✉❝❤ t❤❛t (1 + Crε) N(r) ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✇❤❡r❡

  • N(r) = r d

dr ln max( H(r), r3/2+ε)

  • H(r) =

ˆ r 2ρ

Bρ∩Ω+ |Dw|2dρ = H(r) + O(r3/2+η)

❍❡r❡ t❤❡ ❛❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② ✐s ❝r✉❝✐❛❧ ✐♥ ♦r❞❡r t♦ ❥✉st✐❢② t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✶✾ ✴ ✷✵

slide-48
SLIDE 48

❆❧♠❣r❡♥✲t②♣❡ ♠♦♥♦t♦♥✐❝✐t② ❢♦r♠✉❧❛

■♥ ❣❡♥❡r❛❧✱ ✐❢ H(r) ≤ Cr3/2 ✐t ♠❛❦❡s s❡♥s❡ t♦ ❡①♣❡❝t lim

r→0+ N(r) =

r d

dr

ffl

∂Br∩Ω+ w2

2 ffl

∂Br∩Ω+ w2

≥ 3/2

▲❡♠♠❛

❚❤❡r❡ ❡①✐st ε, η > 0 s✉❝❤ t❤❛t (1 + Crε) N(r) ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✇❤❡r❡

  • N(r) = r d

dr ln max( H(r), r3/2+ε)

  • H(r) =

ˆ r 2ρ

Bρ∩Ω+ |Dw|2dρ = H(r) + O(r3/2+η)

❍❡r❡ t❤❡ ❛❧♠♦st ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② ✐s ❝r✉❝✐❛❧ ✐♥ ♦r❞❡r t♦ ❥✉st✐❢② t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✶✾ ✴ ✷✵

slide-49
SLIDE 49

❇② ❜❧♦✇✐♥❣✲✉♣ t❤❡ s♦❧✉t✐♦♥ ❛♥❞ ✉s✐♥❣ ♦♥❝❡ ❛❣❛✐♥ ❝♦♥✈❡r❣❡ t♦ t❤❡ ❙✐❣♥♦r✐♥✐ ♣r♦❜❧❡♠ ✇❡ r❡❝♦✈❡r t❤❛t lim

r→0+

N(r) ≥ 3/2 ■♥t❡❣r❛t✐♥❣ ✇❡ r❡❝♦✈❡r t❤❡ ❞❡s✐r❡❞ ❜♦✉♥❞

❚❤❛♥❦s✦

✷✵ ✴ ✷✵

slide-50
SLIDE 50

❇② ❜❧♦✇✐♥❣✲✉♣ t❤❡ s♦❧✉t✐♦♥ ❛♥❞ ✉s✐♥❣ ♦♥❝❡ ❛❣❛✐♥ ❝♦♥✈❡r❣❡ t♦ t❤❡ ❙✐❣♥♦r✐♥✐ ♣r♦❜❧❡♠ ✇❡ r❡❝♦✈❡r t❤❛t lim

r→0+

N(r) ≥ 3/2 ■♥t❡❣r❛t✐♥❣ N(r) ∼ r d

dr ln H(r) ✇❡ r❡❝♦✈❡r t❤❡ ❞❡s✐r❡❞ ❜♦✉♥❞

H(r) ≤ Cr3/2

❚❤❛♥❦s✦

✷✵ ✴ ✷✵

slide-51
SLIDE 51

❇② ❜❧♦✇✐♥❣✲✉♣ t❤❡ s♦❧✉t✐♦♥ ❛♥❞ ✉s✐♥❣ ♦♥❝❡ ❛❣❛✐♥ ❝♦♥✈❡r❣❡ t♦ t❤❡ ❙✐❣♥♦r✐♥✐ ♣r♦❜❧❡♠ ✇❡ r❡❝♦✈❡r t❤❛t lim

r→0+

N(r) ≥ 3/2 ■♥t❡❣r❛t✐♥❣ N(r) ∼ r d

dr ln H(r) ✇❡ r❡❝♦✈❡r t❤❡ ❞❡s✐r❡❞ ❜♦✉♥❞

H(r) ≤ Cr3/2

❚❤❛♥❦s✦

✷✵ ✴ ✷✵