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❇♦✉♥❞❛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 ✷✻✲✷✽✱ ❑❚❍

●♦❛❧✿ ❉❡s❝r✐❜❡ ❤♦✇ t❤❡ ❥❡t s❡♣❛r❛t❡s✳ ❆ t❛❧❦ ❛❜♦✉t ❢♦✉♥t❛✐♥s ✶ ✴ ✷✵

❆ t❛❧❦ ❛❜♦✉t ❢♦✉♥t❛✐♥s ●♦❛❧✿ ❉❡s❝r✐❜❡ ❤♦✇ t❤❡ ❥❡t s❡♣❛r❛t❡s✳ ✶ ✴ ✷✵

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

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

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

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

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

❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ✐s r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s ❖♥❡✲♣❤❛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 ❖♥❡✲♣❤❛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 ∈ C 0 , 1 loc ✸ ✴ ✷✵

✐s r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s ❖♥❡✲♣❤❛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 ∈ C 0 , 1 loc ◮ Ω + ❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ✸ ✴ ✷✵

❖♥❡✲♣❤❛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 ∈ C 0 , 1 loc ◮ Ω + ❤❛s ❧♦❝❛❧ ✜♥✐t❡ ♣❡r✐♠❡t❡r ◮ Γ + ✐s C 1 ,α r❡❣✉❧❛r ♣r♦✈✐❞❡❞ ❛ ✢❛t♥❡ss ❤②♣♦t❤❡s✐s ✸ ✴ ✷✵

♦✈❡r ❙❧✐♣ ❝♦♥❞✐t✐♦♥✿ ❍♦✇ ❞♦❡s ❞❡t❛❝❤ ❢r♦♠ ❄ ❏❡ts ❛♥❞ ❝❛✈✐t✐❡s u ≥ 0 s❛t✐s✜❡s � ∆ u = 0 ✐♥ Ω + = { u > 0 } ∩ Ω | Du | = 1 ♦♥ Γ + = ∂ { u > 0 } ∩ Ω ✹ ✴ ✷✵

❏❡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 = { x 2 > 0 } ❍♦✇ ❞♦❡s Γ ❞❡t❛❝❤ ❢r♦♠ Z ❄ ✹ ✴ ✷✵

❘❡❣✉❧❛r✐t② ♦❢ Γ ✉♣ t♦ t❤❡ ✭✜①❡❞✮ ❜♦✉♥❞❛r② ❚❤❡♦r❡♠ ✭❈✲❙❛✈✐♥✮ ▲❡t Ω ⊆ R n ❜❡ ❛ ❞♦♠❛✐♥ ✇✐t❤ ❛ C 1 ,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 ♦♥ Γ 0 = ∂ Ω + ∩ Z ❚❤❡♥ Γ = Γ + ∪ Γ 0 ✐s C 1 , 1 / 2 r❡❣✉❧❛r ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❡✈❡r② x 0 ∈ Γ 0 ✳ ✺ ✴ ✷✵

❘❡❣✉❧❛r✐t② ♦❢ Γ ✉♣ t♦ t❤❡ ✭✜①❡❞✮ ❜♦✉♥❞❛r② ❚❤❡♦r❡♠ ✭❈✲❙❛✈✐♥✮ ▲❡t Ω ⊆ R n ❜❡ ❛ ❞♦♠❛✐♥ ✇✐t❤ ❛ C 1 ,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 ♦♥ Γ 0 = ∂ Ω + ∩ Z ❚❤❡♥ Γ = Γ + ∪ Γ 0 ✐s C 1 , 1 / 2 r❡❣✉❧❛r ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❡✈❡r② x 0 ∈ Γ 0 ✳ ✺ ✴ ✷✵

✭ ¯ F = Γ ✮ ✻ ✴ ✷✵

❚❤❡ ♦♥❡✲♣❤❛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♦✉♥❞ ♦♥ | Du | ≥ 1 ♦♥ Γ 0 ❄ ✼ ✴ ✷✵

❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ ❚❤✐s ✐♠♣❧✐❡s ♦♥ ❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ ♦♥ | Du | ≥ 1 ♦♥ Γ 0 ❄ ❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ ˆ | Du | 2 + χ { u> 0 } min { Ju : u = g ≥ 0 ✐♥ ∂ Ω } Ju = Ω ✼ ✴ ✷✵

❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ ♦♥ | Du | ≥ 1 ♦♥ Γ 0 ❄ ❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ ˆ | Du | 2 + χ { u> 0 } min { Ju : u = g ≥ 0 ✐♥ ∂ Ω } Ju = Ω ❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ u ε ( x + εη ( x )) = u ( x ) ˆ (1 − | Du | 2 ) η · ν + o ( ε ) Ju ε = Ju + ε Γ ❚❤✐s ✐♠♣❧✐❡s | Du | = 1 ♦♥ Γ + ✼ ✴ ✷✵

| Du | ≥ 1 ♦♥ Γ 0 ❄ ❚❤❡ ♦♥❡✲♣❤❛s❡ ♣r♦❜❧❡♠ ❛r✐s❡s ❛s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥ ♦❢ ˆ | Du | 2 + χ { u> 0 } min { Ju : u = g ≥ 0 ✐♥ ∂ Ω } Ju = Ω ❋♦r ❛ ❞♦♠❛✐♥ ✈❛r✐❛t✐♦♥ u ε ( x + εη ( x )) = u ( x ) ˆ (1 − | Du | 2 ) η · ν + o ( ε ) Ju ε = Ju + ε Γ ❚❤✐s ✐♠♣❧✐❡s | Du | = 1 ♦♥ Γ + ❲❡ ❛r❡ ❛❧s♦ ❛❧❧♦✇ t♦ ♣❡r❢♦r♠ ✐♥✇❛r❞ ❞❡❢♦r♠❛t✐♦♥s ❛r♦✉♥❞ Z | Du | ≥ 1 ♦♥ Γ 0 ✼ ✴ ✷✵

❆❧s♦ ✐♥ t❤✐s s❡tt✐♥❣ ✇❡ r❡❝♦✈❡r ♦♥ | Du | ≥ 1 ♦♥ Γ 0 ❄ ❙♦❧✉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 ✳ ✽ ✴ ✷✵

| Du | ≥ 1 ♦♥ Γ 0 ❄ ❙♦❧✉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 ♦♥ Γ 0 ✽ ✴ ✷✵