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  1. ❚r❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s ✇✐t❤ s✐❣♥✲❝❤❛♥❣✐♥❣ ❝♦❡✣❝✐❡♥ts P■❈❖❋ ✷✵✶✷ ❆✳✲❙✳ ❇♦♥♥❡t✲❇❡♥ ❉❤✐❛ † ✱ ▲✳ ❈❤❡s♥❡❧ † ✱ P✳ ❈✐❛r❧❡t † ✱ ❍✳ ❍❛❞❞❛r ‡ † P❖❡♠s t❡❛♠✱ ❊♥st❛✱ P❛r✐s✱ ❋r❛♥❝❡ ‡ ❉❡❋■ t❡❛♠✱ ➱❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ P❛❧❛✐s❡❛✉✱ ❋r❛♥❝❡ ➱❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ P❛❧❛✐s❡❛✉✱ ❋r❛♥❝❡✱ ❆♣r✐❧ ✷✕✹✱ ✷✵✶✷ ✶ ✴ ✶✺

  2. ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ ✌ ✌ ❞✐✈ ✐♥ ❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ◮ ❛♥❞ n ✮ ✐♥ R 2 ✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ D A � = Id ✱ n � = 1 ✷ ✴ ✶✺

  3. ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ ✌ ✌ ❞✐✈ ✐♥ ❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ◮ ❛♥❞ n ✮ ✐♥ R 2 ✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ◮ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ν D A � = Id ✱ n � = 1 ν ✷ ✴ ✶✺

  4. ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ ✌ ❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ◮ ❛♥❞ n ✮ ✐♥ R 2 ✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ◮ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ν ❞✐✈ ( A ∇ u ) + k 2 nu ✐♥ D = 0 D A � = Id ✱ n � = 1 ν ✷ ✴ ✶✺

  5. ❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ◮ ❛♥❞ n ✮ ✐♥ R 2 ✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ◮ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ν ❞✐✈ ( A ∇ u ) + k 2 nu ✐♥ D = 0 D ∆ w + k 2 w ✐♥ D = 0 A � = Id ✱ n � = 1 ν ✷ ✴ ✶✺

  6. ❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ◮ ❛♥❞ n ✮ ✐♥ R 2 ✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ◮ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ν ❞✐✈ ( A ∇ u ) + k 2 nu ✐♥ D = 0 D ∆ w + k 2 w ✐♥ D = 0 A � = Id ✱ n � = 1 ν ♦♥ ∂D u − w = 0 ♦♥ ∂D. ν · A ∇ u − ν · ∇ w = 0 ❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ∂D ✷ ✴ ✶✺

  7. ❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ◮ ❛♥❞ n ✮ ✐♥ R 2 ✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ◮ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ❋✐♥❞ ( u, w ) ∈ H 1 ( D ) × H 1 ( D ) s✉❝❤ t❤❛t✿ ν ❞✐✈ ( A ∇ u ) + k 2 nu ✐♥ D = 0 D ∆ w + k 2 w ✐♥ D = 0 A � = Id ✱ n � = 1 ν ♦♥ ∂D u − w = 0 ♦♥ ∂D. ν · A ∇ u − ν · ∇ w = 0 ❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ∂D ✷ ✴ ✶✺

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