Regularidade e estimativas geométricas para mínimos de funcionais descontínuos e singulares

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❈❯❘❙❖ ❉❊ 01❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❘❆■▼❯◆❉❖ ❆▲❱❊❙ ▲❊■❚➹❖ ❏/◆■❖❘

❘❊●❯▲❆❘■❉❆❉❊ ❊ ❊❙❚■▼❆❚■❱❆❙ ●❊❖▼➱❚❘■❈❆❙ 3❆❘❆

3❘❖❇▲❊▼❆❙ ❱❆❘■❆❈■❖◆❆■❙ ❉❊❙❈❖◆❚❮◆❯❖❙ ❊

❙■◆●❯▲❆❘❊❙

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❘❆■▼❯◆❉❖ ❆▲❱❊❙ ▲❊■❚➹❖ ❏/◆■❖❘

❘❊●❯▲❆❘■❉❆❉❊ ❊ ❊❙❚■▼❆❚■❱❆❙ ●❊❖▼➱❚❘■❈❆❙ 3❆❘❆

3❘❖❇▲❊▼❆❙ ❱❆❘■❆❈■❖◆❆■❙ ❉❊❙❈❖◆❚❮◆❯❖❙ ❊

❙■◆●❯▲❆❘❊❙

❚❡"❡ "✉❜♠❡&✐❞❛ * ❈♦♦-❞❡♥❛/0♦ ❞♦ ❈✉-"♦ ❞❡ 12"✲❣-❛❞✉❛/0♦ ❡♠ ▼❛&❡♠6&✐❝❛ ❞❛ ❯♥✐✈❡-"✐❞❛❞❡ ❋❡❞❡-❛❧ ❞♦ ❈❡❛-6✱ ❝♦♠♦ -❡=✉✐"✐&♦ ♣❛-❝✐❛❧ ♣❛-❛ ♦❜&❡♥/0♦ ❞♦ ❣-❛✉ ❞❡ ❉♦✉&♦- ❡♠ ▼❛&❡♠6&✐❝❛✳

➪-❡❛ ❞❡ ❝♦♥❝❡♥&-❛/0♦✿ ❆♥6❧✐"❡ ▼❛&❡♠6&✐❝❛✳ ❖-✐❡♥&❛❞♦-✿

1-♦❢✳ ❉-✳ ❊❞✉❛-❞♦ ❱❛"❝♦♥❝❡❧♦" ❖❧✐✈❡✐-❛ ❚❡✐①❡✐-❛✳

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Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará

Biblioteca do Curso de Matemática

L548r Leitão Júnior, Raimundo Alves

Regularidade e estimativas geométricas para problemas variacionais descontínuos e singulares / Raimundo Alves Leitão Júnior. – 2012.

117 f. : il. color., enc. ; 31 cm

Tese(doutorado) – Universidade Federal do Ceará, Centro de Ciências, Departamento de Matemática, Programa de Pós-Graduação em Matemática, Fortaleza, 2012.

Área de Concentração: Análise Matemática

Orientação: Prof. Dr. Eduardo Vasconcelos Oliveira Teixeira

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❛❣"❛❞❡❝✐♠❡♥)♦+

●♦"#❛%✐❛ ❞❡ ❡①♣%❡""❛% ♠❡✉" ❛❣%❛❞❡❝✐♠❡♥#♦" ❛♦ 0%♦❢✳ ❉%✳ ❊❞✉❛%❞♦ ❱❛"❝♦♥❝❡❧♦" ❖❧✐✈❡✐%❛ ❚❡✐①❡✐%❛ ♣❡❧❛ "✉❛ ♣❛❝✐:♥❝✐❛ ❡ ❡"#;♠✉❧♦ ❝♦♥"#❛♥#❡✳

❆❣%❛❞❡=♦ ❡♥♦%♠❡♠❡♥#❡ > ♠✐♥❤❛ ♠@❡ ▼❛%✐❛ ▼❛❜❧❡ ❋❡✐#♦"❛ ▲❡✐#@♦✱ ❛♦ ♠❡✉ ❢❛❧❡❝✐❞♦ ♣❛✐ ❘❛✐♠✉♥❞♦ ❆❧✈❡" ▲❡✐#@♦✱ > ♠✐♥❤❛ ✐%♠@ ▼❛%✐❛ ❚❡%❡③❛ ❋❡✐#♦③❛ ▲❡✐#@♦✱ ❛♦ ♠❡✉ ❝✉♥❤❛❞♦ ❈♦"♠❡ ◆♦❣✉❡✐%❛ ▼❛✐❛ ❡ > ♠✐♥❤❛ "♦❜%✐♥❤❛ ❆❧❡①"❛♥❞%❛ ❋❡✐#♦③❛ ▲❡✐#@♦ ▼❛✐❛✳ ◆@♦ ♣♦❞❡%✐❛ ❡"J✉❡❝❡% ❛ ♠✐♥❤❛ ✐%♠@ ▼❛%✐❛ ▲✐③❡#❡ ❋❡✐#♦"❛ ▲❡✐#@♦ ❚❡✐①❡✐%❛✱ ♠❡✉ ✐%♠@♦ ❍❡♥%✐J✉❡ ❏♦%❣❡ ❋❡✐#♦"❛ ▲❡✐#@♦ ❡ ♠❡✉ ❝✉♥❤❛❞♦ ❋%❛♥❝✐"❝♦ ❋❡%♥❛♥❞♦ ❚❡✐①❡✐%❛✳

❈♦♠♦ ♥@♦ ♣♦❞❡%✐❛ ❞❡✐①❛% ❞❡ "❡%✱ ❛❣%❛❞❡=♦ ❛♦" ♠❡✉" ❛♠✐❣♦"✿ ❆♥❞❡%"♦♥ ❋❡✐#♦③❛ ▲❡✐#@♦ ▼❛✐❛✱ ❍N❧❛❞✐♦ ❆♥❞%❛❞❡✱ ❏♦❤♥❛#❤❛♥ ❋❡✐#♦"❛ ❚❡✐①❡✐%❛✱ 0❛✉❧♦ ❍❡♥%✐J✉❡ ❘✐❝❛%❞♦ ▼❛✐❛ ❡ 0❛✉❧♦ ❘✐❝❛%❞♦ 0✐♥❤❡✐%♦ ❙❛♠♣❛✐♦ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♥✜❛♥=❛ J✉❡ ♠❡ ❝♦♥❝❡❞❡%❛♠ ❞✉%❛♥#❡ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞:♠✐❝❛✳

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❡"✉♠♦

❊!"❡ "$❛❜❛❧❤♦ * ❝♦♥!"✐"✉/❞♦ ❞❡ ❞✉❛! ♣❛$"❡!✿ ◆❛ ♣$✐♠❡✐$❛ ♣❛$"❡ ❡!"✉❞❛♠♦! ♠/♥✐♠♦! ♥5♦✲♥❡❣❛"✐✈♦! ❞❡ ❢✉♥❝✐♦♥❛✐! ❡❧/♣"✐❝♦! ❞❡❣❡♥❡$❛❞♦!✱ RF(X, u,∇u)dX →min✱ ♣❛$❛ ♥;❝❧❡♦!

✈❛$✐❛❝✐♦♥❛✐! F <✉❡ !5♦ ❞❡!❝♦♥"/♥✉♦! ❡♠ u ❝♦♠ ❞❡!❝♦♥"✐♥✉✐❞❛❞❡ ❞❡ ♦$❞❡♠ ∼ χ{u>0}✳

❆ ❡<✉❛?5♦ ❞❡ ❊✉❧❡$✲▲❛❣$❛♥❣❡ * ❣♦✈❡$♥❛❞❛ ♣♦$ ✉♠❛ ❡<✉❛?5♦ ❡❧/♣"✐❝❛ ❞❡❣❡♥❡$❛❞❛ ❡ ♥5♦✲❤♦♠♦❣A♥❡❛✱ ❝♦♠ ❢$♦♥"❡✐$❛ ❧✐✈$❡ ❡♥"$❡ ❛! ❢❛!❡! ♣♦!✐"✐✈❛ ❡ ③❡$♦ ❞♦ ♠/♥✐♠♦✳ ▼♦!"$❛$❡♠♦! ❡!"✐♠❛"✐✈❛ ❣$❛❞✐❡♥"❡ D"✐♠❛ ❡ ♥5♦✲❞❡❣❡♥❡$❡!❝A♥❝✐❛ ❞♦ ♠/♥✐♠♦✳ ❚❛♠❜*♠ "$❛"❛$❡♠♦! ❞❡ ♣$♦♣$✐❡❞❛❞❡! ❞❡ $❡❣✉❧❛$✐❞❛❞❡ ❢$❛❝❛! ❡ ❢♦$"❡! ❞❛ ❢$♦♥"❡✐$❛ ❧✐✈$❡✳ F$♦✈❛$❡♠♦! <✉❡ ♦ ❝♦♥❥✉♥"♦ {u > 0} "❡♠ ❧♦❝❛❧♠❡♥"❡ ♣❡$/♠❡"$♦ ✜♥✐"♦ ❡ <✉❡ ❛ ❢$♦♥"❡✐$❛ ❧✐✈$❡ $❡❞✉③✐❞❛ ∂red{u > 0}

"❡♠ ♠❡❞✐❞❛ Hn−1_{✲"♦"❛❧✳ F❛$❛ ♣$♦❜❧❡♠❛! ♠❛✐! ❡!♣❡❝/✜❝♦! <✉❡ ❛♣❛$❡❝❡♠ ❡♠ ❥❡" ✢♦✇&✱} ♣$♦✈❛$❡♠♦! <✉❡ ❛ ❢$♦♥"❡✐$❛ ❧✐✈$❡ $❡❞✉③✐❞❛ * ❧♦❝❛❧♠❡♥"❡ ♦ ❣$I✜❝♦ ❞❡ ✉♠❛ ❢✉♥?5♦ C1,γ_✳

◆❛ !❡❣✉♥❞❛ ♣❛$"❡ ❞♦ "$❛❜❛❧❤♦ ❢♦$♥❡❝❡♠♦! ✉♠❛ ❞❡!❝$✐?5♦ ❜❛!"❛♥"❡ ❝♦♠♣❧❡"❛ ❞❛ "❡♦$✐❛ ❞❡ $❡❣✉❧❛$✐❞❛❞❡ D"✐♠❛ ♣❛$❛ ✉♠❛ ❢❛♠/❧✐❛ ❞❡ ♣$♦❜❧❡♠❛! ❞❡ ❢$♦♥"❡✐$❛ ❧✐✈$❡ ❞❡ ❞✉❛! ❢❛!❡!✱ ❤❡"❡$♦❣A♥❡♦!✱ Jγ →min✱ ❣♦✈❡$♥❛❞♦! ♣♦$ ♦♣❡$❛❞♦$❡! ❡❧/♣"✐❝♦! ❞❡❣❡♥❡$❛❞♦! ❡ ♥5♦✲❧✐♥❡❛$❡!✳ ■♥❝❧✉/❞♦! ❡♠ "❛❧ ❢❛♠/❧✐❛ ❡!"5♦ ♦! ♣$♦❜❧❡♠❛! ❞❡ ❥❡" ✢♦✇& ❤❡"❡$♦❣A♥❡♦! ❡ ♦! ♣$♦❜❧❡♠❛! ❞❡ ❝❛✈✐❞❛❞❡! ❞♦ "✐♣♦ F$❛♥❞"❧✲❇❛"❝❤❡❧♦$✱ γ = 0❀ ❡<✉❛?M❡! ❡❧/♣"✐❝❛! ❞❡❣❡♥❡$❛❞❛! !✐♥❣✉❧❛$❡!

❡ !✐!"❡♠❛! ❞♦ "✐♣♦ ♦❜!"I❝✉❧♦ γ = 1✳ ❱❡$!M❡! ❧✐♥❡❛$❡! ❞❡!"❡! ♣$♦❜❧❡♠❛! "A♠ !✐❞♦ ♦❜❥❡"♦

❞❡ ✐♥"❡♥!❛ ♣❡!<✉✐!❛ ♥❛! ;❧"✐♠❛! <✉❛"$♦ ❞*❝❛❞❛! ♦✉ ♠❛✐!✳ ❆! ❝♦♥"$❛♣❛$"✐❞❛! ♥5♦✲❧✐♥❡❛$❡! "$❛"❛❞❛! ♥❡!"❡ "$❛❜❛❧❤♦ ✐♥"$♦❞✉③❡♠ ♥♦✈❛! ❡ ❝♦♥!✐❞❡$I✈❡✐! ❞✐✜❝✉❧❞❛❞❡!✱ ♣♦✐! ❛ ♠❛✐♦$✐❛ ❞❛! "❡♦$✐❛! ❞❡!❡♥✈♦❧✈✐❞❛! ❛♥"❡$✐♦$♠❡♥"❡✱ "❛✐! ❝♦♠♦ ❢D$♠✉❧❛! ❞❡ ♠♦♥♦"♦♥✐❝✐❞❛❞❡ ❡ ❞❡ '✉❛&❡ ♠♦♥♦"♦♥✐❝✐❞❛❞❡✱ ♥5♦ ❡!"5♦ ❞✐!♣♦♥/✈❡✐!✳ ❈♦♥"✉❞♦✱ ❛! !♦❧✉?M❡! ✐♥♦✈❛❞♦$❛! ❞❡!❡♥✈♦❧✈✐❞❛! ♥❡!"❡ "$❛❜❛❧❤♦ ❢♦$♥❡❝❡♠ ♥♦✈❛! $❡!♣♦!"❛! ♠❡!♠♦ ♥♦ ❝♦♥"❡①"♦ ❝❧I!!✐❝♦ ❞❡ ❡<✉❛?M❡! ❧✐♥❡❛$❡! ❡ ♥5♦✲❞❡❣❡♥❡$❛❞❛!✳

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❛❜"#$❛❝#

❚❤✐# ✇♦&❦ ❝♦♥#✐#*# ♦❢ *✇♦ ♣❛&*#✿ ■♥ *❤❡ ✜&#* ♣❛&* ✇❡ #*✉❞② ♥♦♥♥❡❣❛*✐✈❡ ♠✐♥✐♠✐③❡&# ♦❢ ❣❡♥❡&❛❧ ❞❡❣❡♥❡&❛*❡ ❡❧❧✐♣*✐❝ ❢✉♥❝*✐♦♥❛❧#✱ RF(X, u,∇u)dX→min✱ ❢♦& ✈❛&✐❛*✐♦♥❛❧ ❦❡&♥❡❧#

F *❤❛* ❛&❡ ❞✐#❝♦♥*✐♥✉♦✉# ✐♥ u ✇✐*❤ ❞✐#❝♦♥*✐♥✉✐*② ♦❢ ♦&❞❡& ∼ χ{u>0}✳ ❚❤❡ ❊✉❧❡&✲▲❛❣&❛♥❣❡

❡?✉❛*✐♦♥ ✐# *❤❡&❡❢♦&❡ ❣♦✈❡&♥❡❞ ❜② ❛ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉#✱ ❞❡❣❡♥❡&❛*❡ ❡❧❧✐♣*✐❝ ❡?✉❛*✐♦♥ ✇✐*❤ ❢&❡❡ ❜♦✉♥❞❛&② ❜❡*✇❡❡♥ *❤❡ ♣♦#✐*✐✈❡ ❛♥❞ *❤❡ ③❡&♦ ♣❤❛#❡# ♦❢ *❤❡ ♠✐♥✐♠✐③❡&✳ ❲❡ #❤♦✇ ♦♣*✐♠❛❧ ❣&❛❞✐❡♥* ❡#*✐♠❛*❡ ❛♥❞ ♥♦♥❞❡❣❡♥❡&❛❝② ♦❢ ♠✐♥✐♠❛✳ ❲❡ ❛❧#♦ ❛❞❞&❡## ✇❡❛❦ ❛♥❞ #*&♦♥❣ &❡❣✉❧❛&✐*② ♣&♦♣❡&*✐❡# ♦❢ ❢&❡❡ ❜♦✉♥❞❛&②✳ ❲❡ #❤♦✇ *❤❡ #❡* {u > 0} ❤❛# ❧♦❝❛❧❧② ✜♥✐*❡ ♣❡&✐♠❡*❡& ❛♥❞ *❤❛* *❤❡ &❡❞✉❝❡❞ ❢&❡❡ ❜♦✉♥❞❛&②✱ ∂_❡❞{u > 0}✱ ❤❛# Hn−1_{✲*♦*❛❧ ♠❡❛#✉&❡✳ ❋♦&} ♠♦&❡ #♣❡❝✐✜❝ ♣&♦❜❧❡♠# *❤❛* ❛&✐#❡ ✐♥ ❥❡* ✢♦✇#✱ ✇❡ #❤♦✇ *❤❡ &❡❞✉❝❡❞ ❢&❡❡ ❜♦✉♥❞❛&② ✐# ❧♦❝❛❧❧② *❤❡ ❣&❛♣❤ ♦❢ ❛ C1,γ _{❢✉♥❝*✐♦♥✳}

■♥ *❤❡ #❡❝♦♥❞ ♣❛&* ♦❢ ✇♦&❦ ✇❡ ♣&♦✈✐❞❡ ❛ &❛*❤❡& ❝♦♠♣❧❡*❡ ❞❡#❝&✐♣*✐♦♥ ♦❢ *❤❡ #❤❛&♣ &❡❣✉❧❛&✐*② *❤❡♦&② ❢♦& ❛ ❢❛♠✐❧② ♦❢ ❤❡*❡&♦❣❡♥❡♦✉#✱ *✇♦✲♣❤❛#❡ ✈❛&✐❛*✐♦♥❛❧ ❢&❡❡ ❜♦✉♥❞❛&② ♣&♦❜❧❡♠#✱ Jγ →min✱ &✉❧❡❞ ❜② ♥♦♥❧✐♥❡❛&✱ ❞❡❣❡♥❡&❛*❡ ❡❧❧✐♣*✐❝ ♦♣❡&❛*♦&#✳ ■♥❝❧✉❞❡❞ ✐♥ #✉❝❤ ❢❛♠✐❧② ❛&❡ ❤❡*❡&♦❣❡♥❡♦✉# ❥❡*# ❛♥❞ ❝❛✈✐*✐❡# ♣&♦❜❧❡♠# ♦❢ E&❛♥❞*❧✲❇❛*❝❤❡❧♦& *②♣❡✱ γ = 0❀

#✐♥❣✉❧❛& ❞❡❣❡♥❡&❛*❡ ❡❧❧✐♣*✐❝ ❡?✉❛*✐♦♥# ❛♥❞ ♦❜#*❛❝❧❡ *②♣❡ #✐#*❡♠#✱ γ=1✳ ▲✐♥❡❛& ✈❡&#✐♦♥# ♦❢

*❤❡#❡ ♣&♦❜❧❡♠# ❤❛✈❡ ❜❡❡♥ #✉❜❥❡❝*# ♦❢ ✐♥*❡♥#❡ &❡#❡❛&❝❤ ❢♦& *❤❡ ♣❛#* ❢♦✉& ❞❡❝❛❞❡# ♦& #♦✳ ❚❤❡ ♥♦♥❧✐♥❡❛& ❝♦✉♥*❡&♣❛&*# *&❡❛*❡❞ ✐♥ *❤✐# ♣&❡#❡♥* ✇♦&❦ ✐♥*&♦❞✉❝❡ #✉❜#*❛♥*✐❛❧ ♥❡✇ ❞✐✣❝✉❧*✐❡# #✐♥❝❡ *❤❡ ♠♦#* ♦❢ *❤❡ ❝❧❛##✐❝❛❧ *❤❡♦&✐❡# ❞❡✈❡❧♦♣❡❞ ❡❛&❧✐❡&✱ #✉❝❤ *❤❛* ❛# ♠♦♥♦*♦♥✐❝✐*② ❛♥❞ ❛❧♠♦$% ♠♦♥♦*♦♥✐❝✐*② ❢♦&♠✉❧❛❡✱ ❛&❡ ♥♦ ❧♦♥❣❡& ❛✈❛✐❧❛❜❧❡✳ ◆♦♥❡*❤❡❧❡##✱ *❤❡ ✐♥♥♦✈❛*✐✈❡ #♦❧✉*✐♦♥# ❞❡#✐❣♥❡❞ ✐♥ *❤✐# ✇♦&❦ ♣&♦✈✐❞❡ ♥❡✇ ❛♥#✇❡&# ❡✈❡♥ ✐♥ *❤❡ ❝❧❛##✐❝❛❧ ❝♦♥*❡①* ♦❢ ❧✐♥❡❛&✱ ♥♦♥❞❡❣❡♥❡&❛*❡ ❡?✉❛*✐♦♥#✳

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✉♠#$✐♦

✶ ■◆❚❘❖❉❯➬➹❖ ✶

✶✳✶ ❘❡❣✉❧❛(✐❞❛❞❡ ❡ ❡+,✐♠❛,✐✈❛+ ❣❡♦♠0,(✐❝❛+ ♣❛(❛ ♠3♥✐♠♦+ ❞❡ ❢✉♥❝✐♦♥❛✐+ ❞❡+❝♦♥,3♥✉♦+ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❘❡❣✉❧❛(✐❞❛❞❡ ♣❛(❛ ♣(♦❜❧❡♠❛+ ❞❡ ❢(♦♥,❡✐(❛ ❧✐✈(❡ ❞❡ ❞✉❛+ ❢❛+❡+ ❡ ❞❡❣❡♥❡(❛❞♦+ ✹

✷ +❘❊▲■▼■◆❆❘❊❙ ✶✵

✷✳✶ ❋❡((❛♠❡♥,❛+ ❜:+✐❝❛+ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✶✳✶ ❙❡♠✐❝♦♥,✐♥✉✐❞❛❞❡ ✐♥❢❡(✐♦( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✶✳✷ =(♦❜❧❡♠❛ ❞❡ ♦❜+,:❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✸ ▲❡♠❛ ❞❡ ♠♦♥♦,♦♥✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✹ ▲✐♠✐,❛@A♦ L∞

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✺ ❉❡+✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛❝❝✐♦♣♣♦❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ❋❡((❛♠❡♥,❛+ ♣(✐♥❝✐♣❛✐+ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳✶ ❉❡+✐❣✉❛❧❞❛❞❡ ❞❡ ❍❛(♥❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳✷ ❘❡❣✉❧❛(✐❞❛❞❡ C1,α ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✷✳✸ ❈♦♠♣❛(❛@A♦ ✐♥,❡❣(❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳✹ ❈♦♥✈❡(❣J♥❝✐❛ K✉❛+❡ +❡♠♣(❡ ❞♦ ❣(❛❞✐❡♥,❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✺ ❙♦❧✉@L❡+ ♥♦ +❡♥,✐❞♦ ❞❛ ✈✐+❝♦+✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✻ ❚❡♦(✐❛ ❣❡♦♠0,(✐❝❛ ❞❛ ♠❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✸ ❘❊●❯▲❆❘■❉❆❉❊ ❊ ❊❙❚■▼❆❚■❱❆❙ ●❊❖▼➱❚❘■❈❆❙ +❆❘❆

▼❮◆■▼❖❙ ❉❊ ❋❯◆❈■❖◆❆■❙ ❉❊❙❈❖◆❚❮◆❯❖❙ ✷✹

✸✳✶ ❊①✐+,J♥❝✐❛ ❡ ❝♦♥,✐♥✉✐❞❛❞❡ ❞❡ ♠3♥✐♠♦+ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✷ ❊+,✐♠❛,✐✈❛ ❣(❛❞✐❡♥,❡ P,✐♠❛ ❡ ♥A♦✲❞❡❣❡♥❡(❡+❝J♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✸ ❊+,✐♠❛,✐✈❛+ ❞❡ ❍❛✉+❞♦(✛ ❞❛ ❢(♦♥,❡✐(❛ ❧✐✈(❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✹ =(♦❜❧❡♠❛+ ❧✐♥❡❛(❡+ ❞❡ ❥❡" ✢♦✇& ❡ ❛ +✉❛✈✐❞❛❞❡ ❞❛ ❢(♦♥,❡✐(❛ ❧✐✈(❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

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✹ ❘❊●❯▲❆❘■❉❆❉❊ )❆❘❆ )❘❖❇▲❊▼❆❙ ❉❊ ❋❘❖◆❚❊■❘❆ ▲■❱❘❊ ❉❊

❉❯❆❙ ❋❆❙❊❙ ❊ ❉❊●❊◆❊❘❆❉❖❙ ✻✻

✹✳✶ ❊①✐&'(♥❝✐❛ ❡ ❧✐♠✐'❛/0♦ L∞ ❞❡ ♠3♥✐♠♦& ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✷ ❊&'✐♠❛'✐✈❛& 7'✐♠❛& C1,α ♣❛9❛ ♠3♥✐♠♦& ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

✹✳✸ ❊&'✐♠❛'✐✈❛& ❧♦❣✲▲✐♣&❝❤✐'③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✹ ▲✐♠✐'❛/B❡& ✐♥❢❡9✐♦9❡& ❞♦ ❣9❛❞✐❡♥'❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✺ ❊&'❛❜✐❧✐❞❛❞❡ ❞❡ ♣9♦❜❧❡♠❛& ❞❡ ❢9♦♥'❡✐9❛ ❧✐✈9❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

❘❊❋❊❘✃◆❈■❆❙ ✽✺

❆)✃◆❉■❈❊❙ ✾✶

✹✳✻ ❋❛'♦& ❝♦♥❤❡❝✐❞♦& ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✹✳✼ ❆❧❣✉♥& ♣9♦❥❡'♦& ❞❡ ♣❡&K✉✐&❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✹✳✼✳✶ ❚❡♦9✐❛ ❞❡ 9❡❣✉❧❛9✐❞❛❞❡ ❣❡♦♠M'9✐❝❛ ♣❛9❛ ❛ ❢9♦♥'❡✐9❛ ❧✐✈9❡ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✹✳✼✳✷ ❘❡❣✉❧❛9✐❞❛❞❡ ♣❛9❛ ❡K✉❛/B❡& ♣❛9❛❜7❧✐❝❛& ❞✉♣❧❛♠❡♥'❡ ♥0♦✲❧✐♥❡❛9❡& ✳ ✳ ✾✺ ✹✳✼✳✸ O❡9'✉❜❛/0♦ &✐♥❣✉❧❛9 ♣❛9❛ ❡K✉❛/B❡& ♣❛9❛❜7❧✐❝❛&✱ ❞❡❣❡♥❡9❛❞❛& ❡

(11)

❛❜❡❧❛ ❞❡ &'♠❜♦❧♦&

◮ Rn ✿ ❊"♣❛%♦ ❡✉❝❧✐❞✐❛♥♦ ❞❡ ❞✐♠❡♥"/♦ _n

◮ Ω ✿ ❉♦♠1♥✐♦✱ ✐"3♦ 4✱ ✉♠ ❝♦♥❥✉♥3♦ ❛❜❡73♦ ❡ ❝♦♥❡①♦ ❞❡ Rn

◮ Ω′ ✿ ❋❡❝❤♦ ❞❡ Ω′ ❡♠ Rn

◮ Ω′ ⋐Ω✿ Ω′ 4 ❝♦♠♣❛❝3♦ ❡ Ω′ ⊂Ω

◮ ε≪1 ✭ε≫1✮✿ ε 4 "✉✜❝✐❡♥3❡♠❡♥3❡ ♣❡>✉❡♥♦ ✭"✉✜❝✐❡♥3❡♠❡♥3❡ ❣7❛♥❞❡✮ ◮ |E|✿ ▼❡❞✐❞❛ n✲❞✐♠❡♥"✐♦♥❛❧ ❞❡ ▲❡❜❡"❣✉❡ ❞♦ ❝♦♥❥✉♥3♦ E

◮ Hn−1✿ ▼❡❞✐❞❛ (n−1)✲❞✐♠❡♥"✐♦♥❛❧ ❞❡ ❍❛✉"❞♦7✛ ◮ ❞✐"3(X, E) ✿ inf{|X−Z|;Z∈E}

◮ N_δ(E)✿ {X∈Rn:❞✐"3(X, E)< δ}✱E⊂Rn ◮ B_r(X₀)✿ {X∈Rn;|X−X₀|< r}

◮ h., .i✿ E7♦❞✉3♦ ❡"❝❛❧❛7 ❞❡ Rn ◮ u+ ✭u−✮✿ max{u, 0} ✭−min{u, 0}✮

◮ χE✿ ❋✉♥%/♦ ❝❛7❛❝3❡71"3✐❝❛ ❞♦ ❝♦♥❥✉♥3♦ E ◮ f=o(g)>❞♦ x _→x0✿ lim_x→_x₀

|f(x)| |g(x)| =0

◮ Z

Br(x0)

u(x)dx✿ _|B 1

r(x0)| R

Br(x0)u(x)dx

◮ ∂u

∂xi✱ ∂iu✿ i✲4"✐♠❛ ❞❡7✐✈❛❞❛ ♣❛7❝✐❛❧ ❞❛ ❢✉♥%/♦ u:

Rn

→R

(12)

◮ Ck(Ω)✿ ❋✉♥$%❡' k✲✈❡③❡' ❝♦♥-✐♥✉❛♠❡♥-❡ ❞✐❢❡3❡♥❝✐4✈❡✐' ❡♠ Ω

◮ C∞

0 (Ω)✿ ❋✉♥$%❡' u∈C ∞

(Ω) ❝♦♠ '✉♣♦3-❡ ❝♦♠♣❛❝-♦ ❡♠ Ω

◮ C0,α

loc(Ω)✿ ❋✉♥$%❡' ❧♦❝❛❧♠❡♥-❡ α✲❍8❧❞❡3 ❝♦♥-9♥✉❛' ❡♠ Ω✱ 0 < α≤1

◮ Ck,α

loc(Ω)✿ ❋✉♥$%❡' u∈Ck(Ω) ❝♦♠ ❞❡3✐✈❛❞❛' ❞❡ ♦3❞❡♠ k ❡♠ C0,αloc(Ω)

◮ Lp(Ω)✿ ❋✉♥$%❡' ;✉❡ ♣♦''✉❡♠ ❛ p✲<'✐♠❛ ♣♦-=♥❝✐❛ ✐♥-❡❣34✈❡❧ ❡♠ Ω ◮ k k_Lp(D)✿ ◆♦3♠❛ ❞♦ ❡'♣❛$♦ Lp(D)✱D⊂Ω

◮ W1,p(D)✿ ❋✉♥$%❡' ❡♠ Lp(D) ❝✉❥❛' ❛' ❞❡3✐✈❛❞❛' ❞❡ ♣3✐♠❡✐3❛ ♦3❞❡♠ ❞✐'-3✐❜✉❝✐♦♥❛✐'

❡'-B♦ ❡♠ Lp₍_D₎

◮ kuk_W1,p₍_D₎✿ ♥♦3♠❛ W1,p ❞❡ u♥✉♠ ❝♦♥❥✉♥-♦ D⊂Ω ◮ W1,p

0 (D)✿ ❋❡❝❤♦ ❞❡ C ∞

0 (D) ❝♦♠ 3❡'♣❡✐-♦ ❛ ♥♦3♠❛ W1,p

(13)

✶

❈❛♣#$✉❧♦

1

■◆❚❘❖❉❯➬➹❖

❈♦♥$❡*❞♦

✶✳✶ ❘❡❣✉❧❛(✐❞❛❞❡ ❡ ❡+,✐♠❛,✐✈❛+ ❣❡♦♠0,(✐❝❛+ ♣❛(❛ ♠3♥✐♠♦+ ❞❡ ❢✉♥❝✐♦♥❛✐+ ❞❡+❝♦♥,3♥✉♦+ ✶ ✶✳✷ ❘❡❣✉❧❛(✐❞❛❞❡ ♣❛(❛ ♣(♦❜❧❡♠❛+ ❞❡ ❢(♦♥,❡✐(❛ ❧✐✈(❡ ❞❡ ❞✉❛+ ❢❛+❡+ ❡ ❞❡❣❡♥❡(❛❞♦+ ✳ ✳ ✳ ✹

✶✳✶ ❘❡❣✉❧❛(✐❞❛❞❡ ❡ ❡+,✐♠❛,✐✈❛+ ❣❡♦♠0,(✐❝❛+ ♣❛(❛

♠3♥✐♠♦+ ❞❡ ❢✉♥❝✐♦♥❛✐+ ❞❡+❝♦♥,3♥✉♦+

❉❛❞♦ ✉♠ ❞♦♠(♥✐♦ ❧✐♠✐,❛❞♦ -✉❛✈❡ Ω ⊂ Rn _{❡ ✉♠❛ ❢✉♥12♦ ❧✐♠✐,❛❞❛ ♥2♦✲♥❡❣❛,✐✈❛} _φ _∈

W1,p₍_Ω₎_✱ ₂ _≤ _{p < n}_{✱ ❡-,✉❞❛6❡♠♦- 6❡❣✉❧❛6✐❞❛❞❡ ❡ ♣6♦♣6✐❡❞❛❞❡- ❣❡♦♠8,6✐❝❛- ❞❛ -♦❧✉12♦} ♣❛6❛ ♦ -❡❣✉✐♥,❡ ♣6♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛12♦

min Z

Ω

F(X, u,∇u)dX:u∈W_φ1,p(Ω)

, ✭✶✳✶✮

♦♥❞❡ W_φ1,p(Ω) ❞❡♥♦,❛ ♦ ❡-♣❛1♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ ,♦❞❛- ❛- ❢✉♥1?❡- ❡♠ Lp₍_Ω₎ _❝♦♠ ❞❡6✐✈❛❞❛-❞✐-,6✐❜✉❝✐♦♥❛✐- ❡♠ Lp₍_Ω₎ _{❡ ,6❛1♦} _φ_{✳ ❖ ♥A❝❧❡♦ ✈❛6✐❛❝✐♦♥❛❧}_F_: _Ω_×_R_×_Rn

→R -❛,✐-❢❛③

❛--❡❣✉✐♥,❡- ❝♦♥❞✐1?❡- ❡-,6✉,✉6❛✐-✿ F(X, u, ξ) =G(X, ξ) +g(X, u) ❡ ✭●✶✮ D❛6❛ ,♦❞♦ ξ∈Rn✱ ❛ ❛♣❧✐❝❛12♦ _X₇

→G(X, ξ)8 ❝♦♥,(♥✉❛✳

✭●✷✮ ❊①✐-,❡ ✉♠❛ ❝♦♥-,❛♥,❡ ♣♦-✐,✐✈❛ 0 < λ ,❛❧ H✉❡✱

λ|ξ|p≤G(X, ξ)≤λ−1|ξ|p.

(14)

✷ ❞✐❢❡%❡♥❝✐(✈❡❧ ❡ +❛-✐+❢❛③

G(X, tξ) =|t|pG(X, ξ), t ∈R_{, ξ}_∈Rn_.

✭●✹✮ ❊①✐+-❡♠ ❝♦♥+-❛♥-❡+ 0 < δ < 1 ❡CA > 0 -❛✐+ 7✉❡

X7_→A(X, ξ)∈Cδ(Ω′), sup

ξ∈Rn

kA(X, ξ)kCδ ≤C_A,

♣❛%❛ -♦❞♦ +✉❜❞♦♠;♥✐♦ Ω′ _⊂_Ω_{✱ ♦♥❞❡}_A₍_{X, ξ}_{) :=}_∇

ξG(X, ξ)✳ ✭❣✶✮ ❆ ❢✉♥AB♦ g C ❞❡✜♥✐❞❛ ♣♦%

g(X, u) =f(X) (u+)m+Qχ{u>0}, 1≤m < p,

♦♥❞❡ f C ♠❡♥+✉%(✈❡❧✱ −K≤ f≤ K✱ ♣❛%❛ ❛❧❣✉♠ K > 0❀ Q C C0,β_{✲❝♦♥-;♥✉❛✱} _{0 < ǫ <}

Q < ǫ−1 _{♣❛%❛ ❛❧❣✉♠} _{ǫ > 0}_✳

❯♠ ♣%♦-H-✐♣♦ ✐♠♣♦%-❛♥-❡ ❞❡ ♥I❝❧❡♦ ✈❛%✐❛❝✐♦♥❛❧ ♣❛%❛ ♠❛♥-❡% ❡♠ ♠❡♥-❡ C

F(X, u, ξ) =|ξ|p−2A(X)ξ·ξ+f(X) (u+)m+Qχ{u>0}, ✭✶✳✷✮ ♣❛%❛ ✉♠❛ ♠❛-%✐③ ♣♦+✐-✐✈❛ ❞❡✜♥✐❞❛ A ❝♦♠ ❝♦❡✜❝✐❡♥-❡+ ❝♦♥-;♥✉♦+✳ ▼♦-✐✈❛AK❡+ ✈✐♥❞❛+

❞♦ ❡+-✉❞♦ ❞❡ ❥❡" ✢♦✇&✱ ♣%♦❜❧❡♠❛+ ❞❡ ❝❛✈✐❞❛❞❡✱ ❡♥-%❡ ♠✉✐-❛+ ♦✉-%❛+ ❛♣❧✐❝❛AK❡+✳ L♦% ❝♦♥✈❡♥✐M♥❝✐❛ ❞❡ ♥♦-❛AB♦✱ ♦ ❢✉♥❝✐♦♥❛❧ 7✉❡ ❛♣❛%❡❝❡ ♥♦ ♣%♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛AB♦ ✭✶✳✶✮ +❡%( ❞❡♥♦-❛❞♦ ♣♦% F : W_φ1,p(Ω)_→R✱ ✐✳❡✳✱ ❞❛7✉✐ ♣♦% ❞✐❛♥-❡

F(u) :=

Z

Ω

F(X, u,∇u)dX.

❚❛♠❜C♠✱ 7✉❛❧7✉❡% ❝♦♥+-❛♥-❡ ♣♦+✐-✐✈❛ C=C(n, p, m, λ, φ, ǫ, K, Ω)7✉❡ ❞❡♣❡♥❞❡ +♦♠❡♥-❡ ❞❛ ❞✐♠❡♥+B♦ ❡ ❞♦+ ♣❛%O♠❡-%♦+ ❝♦♥+-❛♥-❡+ ❞♦ ♣%♦❜❧❡♠❛ +❡%( ❝❤❛♠❛❞♦✱ ❞❛7✉✐ ♣♦% ❞✐❛♥-❡✱ ✉♠❛ ❝♦♥+-❛♥-❡ ✉♥✐✈❡+&❛❧✳

❆ ❝❛%❛❝-❡%;+-✐❝❛ ❝❤❛✈❡ ❞♦ ❢✉♥❝✐♦♥❛❧FC ❛ +✉❛ ❞❡+❝♦♥-✐♥✉✐❞❛❞❡ ❝♦♠ %❡+♣❡✐-♦ ❛u✱ ❛++✐♠ ❛

❜❡♠ ❡+-❛❜❡❧❡❝✐❞❛ -❡♦%✐❛ ❝❧(++✐❝❛ ❞♦ ❈(❧❝✉❧♦ ❞❛+ ❱❛%✐❛AK❡+ ♥B♦ C ❛❞❡7✉❛❞❛ ♣❛%❛ -%❛-❛% -❛✐+ ♣%♦❜❧❡♠❛+✳ ❉❡ ❢❛-♦✱ ♣❛%❛ ✉♠ ♠;♥✐♠♦ ❡①✐+-❡♥-❡ u✱ ♦ ❢✉♥❝✐♦♥❛❧F❛♣%❡+❡♥-❛ ❞❡+❝♦♥-✐♥✉✐❞❛❞❡ ♣❛%❛ ♣❡7✉❡♥❛+ ♣❡%-✉❜❛AK❡+ ♣%H①✐♠❛+ ❞❡ ♣♦♥-♦+ ❞♦✱ ❛ ♣%✐♥❝;♣✐♦ ❞❡+❝♦♥❤❡❝✐❞♦✱ ❝♦♥❥✉♥-♦ {u > 0}✳ ❚❛❧ ❞❡+❝♦♥-✐♥✉✐❞❛❞❡ %❡✢❡-❡ ❡♠ ✉♠❛ ❢❛❧-❛ ❞❡ +✉❛✈✐❞❛❞❡ ❞❡ u❛♦ ❝%✉③❛% ❛ ❢%♦♥-❡✐%❛

❞❛ +✉❛ +✉♣❡%❢;❝✐❡ ❞❡ ♥;✈❡❧ ③❡%♦✳

(15)

✸

A(X) = ■❞ ❡♠ ✭✶✳✷✮✳ ❉❛♥✐❡❧❧✐ ❡ /❡01♦3②❛♥ ❡♠ ❬✶✹❪ ❞❡3❡♥✈♦❧✈❡1❛♠ ❛ 0❡♦1✐❛ ❝♦11❡3♣♦♥❞❡♥0❡ ❞❡ ❆❧0 ❡ ❈❛✛❛1❡❧❧✐ ♣❛1❛ ♦ ♣✲▲❛♣❧❛❝❡✱ ✐✳❡✳✱ f≡0 ❡A(X) = ■❞✳

◆❡30❡ 01❛❜❛❧❤♦ ❡30✉❞❛♠♦3 ♦ ♣1♦❜❧❡♠❛ ✈❛1✐❛❝✐♦♥❛❧ ✭✶✳✶✮ ❡♠ 3✉❛ 0♦0❛❧ ❣❡♥❡1❛❧✐❞❛❞❡✱ ♣1♦✈❛♥❞♦ ❡①✐30G♥❝✐❛✱ 1❡❣✉❧❛1✐❞❛❞❡ ❡ ♣1♦♣1✐❡❞❛❞❡3 ❣❡♦♠H01✐❝❛3 ❞❡ ❝❡10♦3 ♣1♦❜❧❡♠❛3 ❞❡ ❢1♦♥0❡✐1❛ ❧✐✈1❡ ❤❡0❡1♦❣G♥❡♦3 ❡ ❣♦✈❡1♥❛❞♦3 ♣♦1 ❡J✉❛KL❡3 ❡❧M♣0✐❝❛3 ❞❡❣❡♥❡1❛❞❛3✳ ❖3 1❡3✉❧0❛❞♦3 ❞❡30❡ 01❛❜❛❧❤♦ 3O♦ ♥♦✈♦3 ♣❛1❛ ❛ ❡J✉❛KO♦ ❞♦ 0✐♣♦ /♦✐33♦♥ m = 1✳ ❚❛♠❜H♠

01❛③ ♥♦✈♦3 1❡3✉❧0❛❞♦3 ♠❡3♠♦ ♥♦ ❝❛3♦ ❧✐♥❡❛1 p=2✳

◆❛ ❙❡KO♦ ✸✳✶ ♠♦301❛1❡♠♦3 ❛ ❡①✐30G♥❝✐❛ ❞❡ ✉♠ ♠M♥✐♠♦ ♣❛1❛ ♦ ❢✉♥❝✐♦♥❛❧ F✱ 0❛❧ ♠M♥✐♠♦ H ♥O♦✲♥❡❣❛0✐✈♦ ❡ ❝♦♥0M♥✉♦ ❡♠ Ω✳ ❚❛♠❜H♠ ♣1♦✈❛1❡♠♦3 J✉❡ ♥♦ ❝♦♥❥✉♥0♦ ❞❡ ♣♦3✐0✐✈✐❞❛❞❡✱ u 3❛0✐3❢❛③ ❛ ❡J✉❛KO♦ ❞❡ ❊✉❧❡1✲▲❛❣1❛♥❣❡ ❞❡3❡❥❛❞❛✱

❞✐✈(∇ξG(X,∇u)) =mf(X)um−1, ❡♠ {u > 0},

♥♦ 3❡♥0✐❞♦ ❞❛3 ❞✐301✐❜✉✐KL❡3✳ ❊♠ ♣❛0✐❝✉❧❛1✱ uHC1,ǫ _{❡♠ 0❛❧ ❝♦♥❥✉♥0♦ ✭♥❡30❡ ♣♦♥0♦ ❡①✐❣✐♠♦3} J✉❡G(X, ξ)♣♦33✉❛ ❝♦❡✜❝✐❡♥0❡3C1(Ω)✮✳ ❈♦♥0✉❞♦✱ ❞❡✈✐❞♦ ❛ ❞❡3❝♦♥0✐♥✉✐❞❛❞❡ ❞❡F♣1V①✐♠♦ ❞♦3 ♣♦♥0♦3 ❞❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡✱ ∇u 3❛❧0❛ ❞❡ ✈❛❧♦1❡3 ♣♦3✐0✐✈♦3 ❛ ③❡1♦ ❛01❛✈H3 ❞♦ ❝♦♥❥✉♥0♦ ∂{u > 0}✳ /♦1 ✐33♦✱ ❛ 1❡❣✉❧❛1✐❞❛❞❡ V0✐♠❛ ♣❛1❛ ♦ ♠M♥✐♠♦ H ❛ ❝♦♥0✐♥✉✐❞❛❞❡ ▲✐♣3❝❤✐0③✳ ❚❛❧ 1❡3✉❧0❛❞♦ H ❡30❛❜❡❧❡❝✐❞♦ ♥❛ ❙❡KO♦ ✸✳✷✳ /❡❧❛ 1❡❣✉❧❛1✐❞❛❞❡ ▲✐♣3❝❤✐0③✱ ❝♦♥❝❧✉M♠♦3 J✉❡ u

❝1❡3❝❡ ♥♦ ♠W①✐♠♦ ❞❡ ❢♦1♠❛ ❧✐♥❡❛1 ❧♦♥❣❡ ❞❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡✳ ◆♦ ❡♥0❛♥0♦✱ ♣♦1 ❝♦♥3✐❞❡1❛KL❡3 ❞❡ ❡♥❡1❣✐❛✱ ♠♦301❛1❡♠♦3 J✉❡ u ❝1❡3❝❡ ♣1❡❝✐3❛♠❡♥0❡ ❞❡ ❢♦1♠❛ ❧✐♥❡❛1 ❧♦♥❣❡ ❞❡ ∂{u > 0}✳ ❊30❛ H ✉♠❛ ✐♥❢♦1♠❛KO♦ ❣❡♦♠H01✐❝❛ ✐♠♣♦10❛♥0❡ J✉❡ ❞❛1 ❛❝❡33♦ ❛ ❝❛1❛❝0❡1M30✐❝❛3 ✜♥❛3 ♥♦ 3❡♥0✐❞♦ ❞❛ 0❡♦1✐❛ ❣❡♦♠H01✐❝❛ ❞❛ ♠❡❞✐❞❛ ❞❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡✳ ❉❡ ❢❛0♦✱ ♥❛ ❙❡KO♦ ✸✳✸

Λ:= ❞✐✈(∇ξG(X,∇u)) −mf(X)um−1,

❞❡✜♥❡ ✉♠❛ ♠❡❞✐❞❛ ♥O♦✲♥❡❣❛0✐✈❛ 3✉♣♦10❛❞❛ ♥❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡✳ ❚❛♠❜H♠ ♠♦301❛1❡♠♦3 J✉❡ ♦ ❝♦♥❥✉♥0♦ ❞❡ ♣♦3✐0✐✈✐❞❛❞❡ ❞❡ u✱ {u > 0}✱ H ❧♦❝❛❧♠❡♥0❡ ✉♠ ❝♦♥❥✉♥0♦ ❞❡ ♣❡1M♠❡01♦ ✜♥✐0♦✳ ❯♠❛ ♣1♦♣1✐❡❞❛❞❡ ✜♥❛ H ♠♦301❛❞❛✿ ✈❡1✐✜❝❛♠♦3 J✉❡

Hn−1₍_∂_{_{u > 0}_}_∩_B_r(_Z₎₎_∼_rn−1_,

♣❛1❛ J✉❛❧J✉❡1 ❜♦❧❛ Br(Z) ❝❡♥01❛❞❛ ❡♠ ✉♠ ♣♦♥0♦ ❞❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡✳ ❊♠ ♣❛10✐❝✉❧❛1✱ ❝♦♥❝❧✉M♠♦3 J✉❡ ❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡ 1❡❞✉③✐❞❛✱ ∂_❡❞{u > 0}✱ 0❡♠ ♠❡❞✐❞❛ Hn−1_{✲❍❛✉3❞♦1✛ 0♦0❛❧✳} /♦1 [❧0✐♠♦ ♥❛ ❙❡KO♦ ✸✳✹ 01❛0❛♠♦3 ❞❛ 3✉❛✈✐❞❛❞❡ ❞❛ ❢1♦♥0❡✐1❛ ❧✐✈1❡ ✭1❡❞✉③✐❞❛✮ ♣❛1❛ ♦ ♣1♦❜❧❡♠❛ ❞❡ ❝❛✈✐❞❛❞❡ ❤❡0❡1♦❣G♥❡♦ ❡ J✉❛3❡✲❧✐♥❡❛1

  

 

div(A(X)∇u) = mf(X)um−1_, _❡♠ _{_{u > 0}_}

(16)

✹ !"♦✈❛"❡♠♦( )✉❡ ❛ ❢"♦♥-❡✐"❛ ❧✐✈"❡ 0 ✉♠❛ (✉♣❡"❢2❝✐❡ (✉❛✈❡ C1,γ_{✱ ❛ ♠❡♥♦( ❞❡ ✉♠ ♣♦((2✈❡❧} ❝♦♥❥✉♥-♦ ❞❡ ♠❡❞✐❞❛ Hn−1 _{♥✉❧❛✱ ♠♦(-"❛♥❞♦ )✉❡ ✉♠❛ (♦❧✉78♦ ❝❧9((✐❝❛ ❝♦""❡(♣♦♥❞❡ ❛ ✉♠} ♣"♦❜❧❡♠❛ )✉❛(❡✲❧✐♥❡❛" -✐♣♦ ❇❡"♥♦✉❧❧✐✳

✶✳✷ ❘❡❣✉❧❛)✐❞❛❞❡ ♣❛)❛ ♣)♦❜❧❡♠❛0 ❞❡ ❢)♦♥3❡✐)❛ ❧✐✈)❡ ❞❡

❞✉❛0 ❢❛0❡0 ❡ ❞❡❣❡♥❡)❛❞♦0

❙❡❥❛ Ω ⊂Rn ✉♠ ❞♦♠2♥✐♦ ❧✐♠✐-❛❞♦✱ ₂ _≤_{p < n}✱ _f_∈ _Lq₍_Ω₎✱ _q _≥ _n ❡ _ϕ _∈ _W1,p₍_Ω₎_∩

L∞

(Ω)✱ ❝♦♠✱ ❞✐❣❛♠♦(✱ ϕ+ 6= 0✳ ◆❡(-❡ -"❛❜❛❧❤♦ ♦❜-❡"❡♠♦( ❡(-✐♠❛-✐✈❛( ❞❡ "❡❣✉❧❛"✐❞❛❞❡

✐♥-❡"✐♦" B-✐♠❛( ♣❛"❛ ♠2♥✐♠♦( ❞❡ ❢✉♥❝✐♦♥❛✐( ♥8♦✲❞✐❢❡"❡♥❝✐9✈❡✐( ❡ ❤❡-❡"♦❣C♥❡♦(

Jγ(v) :=

Z

Ω

(|∇v|p+Fγ(v) +f(X)·v)dX−_→♠✐♥, ✭✶✳✸✮

❡♥-"❡ ❛( ❢✉♥7H❡( )✉❡ ❝♦♠♣❡-❡♠ v ∈ W₀1,p(Ω) + ϕ✱ 0 ≤ γ ≤ 1 0 ♦ ♣❛"I♠❡-"♦ ❞❡

(✐♥❣✉❧❛"✐❞❛❞❡✱ 0≤λ− < λ+ <∞(8♦ ❡(❝❛❧❛"❡( ❡ ♦ ♣♦-❡♥❝✐❛❧ Fγ 0 ❞❛❞♦ ♣♦"

Fγ(v) := λ+(v+)γ+λ−(v−)γ. ✭✶✳✹✮

❯(✉❛❧♠❡♥-❡✱ v±_:= _max_{_±_{v, 0}_}_{✱ ❡ ♣♦" ❝♦♥✈❡♥78♦✱}

F0(v) := λ+χ{v>0}+λ−χ{v≤0}. ✭✶✳✺✮ ❆ ♥8♦✲❞✐❢❡"❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞♦ ♣♦-❡♥❝✐❛❧Fγ✐♠♣❡❧❡ M ❡)✉❛78♦ ❞❡ ❊✉❧❡"✲▲❛❣"❛♥❣❡ ❛((♦❝✐❛❞❛ ❛ Jγ (❡" (✐♥❣✉❧❛" ❛♦ ❧♦♥❣♦ ❞❛ ❢"♦♥-❡✐"❛ ❧✐✈"❡ ❛ ♣"✐♦"✐ ❞❡(❝♦♥❤❡❝✐❞❛

Fγ := (∂{uγ > 0}∪∂{uγ < 0})∩Ω, ✭✶✳✻✮ ❡♥-"❡ ❛( ❢❛(❡( ♥❡❣❛-✐✈❛ ❡ ♣♦(✐-✐✈❛ ❞❡ ✉♠ ♠2♥✐♠♦✳ ❉❡ ❢❛-♦✱ ✉♠ ♠2♥✐♠♦ (❛-✐(❢❛③✱ ❡♠ ❛❧❣✉♠ (❡♥-✐❞♦ ❢"❛❝♦✱ ❛ (❡❣✉✐♥-❡ ❊❉! (✐♥❣✉❧❛" ❡ p✲❞❡❣❡♥❡"❛❞❛

∆pu=

γ p

"

λ+(u+)γ−1χ{u>0}−λ−(u−)γ−1χ{u≤0}

+ 1

pf(X) ❡♠ Ω, ✭✶✳✼✮

♦♥❞❡ ∆p ❞❡♥♦-❛ ♦ ♦♣❡"❛❞♦" p✲▲❛♣❧❛❝✐❛♥♦✱

∆pu=❞✐✈(|∇u|p−2∇u).

(17)

✺

|∇u+|p−|∇u−|p = 1

p−1(λ+−λ−), ❡♠ {u=0}, ✭✶✳✽✮

❛♦ ❧♦♥❣♦ ❞❛ ❢/♦♥0❡✐/❛ ❧✐✈/❡ ❞♦ ♣/♦❜❧❡♠❛✱ ♦ 6✉❡ 6✉❡❜/❛ ❛ ❝♦♥0✐♥✉✐❞❛❞❡ ❞♦ ❣/❛❞✐❡♥0❡ ❛0/❛✈9: ❞❡ F0✳

❯♠ ♥<♠❡/♦ ❞❡ ♣/♦❜❧❡♠❛: ♠❛0❡♠=0✐❝♦: ✐♠♣♦/0❛♥0❡:✱ ✈✐♥❞♦: ❞❡ ✈=/✐♦: ❝♦♥0❡①0♦: ❞✐❢❡/❡♥0❡:✱ :?♦ ♠♦❞❡❧❛❞♦: ♣♦/ ♠♦❞❡❧♦: ❞❡ ♦0✐♠✐③❛A?♦✱ ♣❛/❛ ♦: 6✉❛✐: ❛ ❡6✉❛A?♦ ✭✶✳✸✮ :❡/✈❡ ❝♦♠♦ ✉♠ ♣/♦0C0✐♣♦ ♣/✐♥❝✐♣❛❧ ❡♠❜❧❡♠=0✐❝♦✳ ❊:0❡ ❢❛0♦ 0❡♠ ❡♥❝♦/❛❥❛❞♦ ✐♥✈❡:0✐❣❛AF❡: ♠❛❝✐A❛:✱ ❡ ✈❡/:F❡: ❧✐♥❡❛/❡:✱ p=2✱ ❞♦ ♣/♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛A?♦ ✭✶✳✸✮ 0❡♠ ❞❡ ❢❛0♦ /❡❝❡❜✐❞♦

✉♠❛ ✐♥0❡♥:❛ ❛0❡♥A?♦ ♥❛: <❧0✐♠❛: 6✉❛0/♦ ❞9❝❛❞❛:✳ ❖ ❝❛:♦ :✉♣❡/✐♦/ γ=1❡:0= /❡❧❛❝✐♦♥❛❞♦ ❛

♣/♦❜❧❡♠❛: ❞❡ ♦❜:0=❝✉❧♦✳ H/♦❜❧❡♠❛: ❞♦ 0✐♣♦ ♦❜:0=❝✉❧♦ ❞❡ ✉♠❛ ❢❛:❡ ❤♦♠♦❣J♥❡♦: ❡ ❧✐♥❡❛/❡: ✐✳❡✳✱ p = 2✱ f(X) ≡ 0 ❡ ϕ ≥ 0 ❢♦/❛♠ 0♦0❛❧♠❡♥0❡ ❡:0✉❞❛❞♦: ♥♦: ❛♥♦: ✼✵ ♣♦/ ❋/❡❤:❡✱

❙0❛♠♣❛❝❝❤✐❛✱ ❑✐♥❡/❧❡❤/❡/✱ ❇/❡③✐:✱ ❈❛✛❛/❡❧❧✐✱ ❡♥0/❡ ♦✉0/♦:✳ ◆❡:0❡ ❝❡♥=/✐♦✱ ❢♦✐ ❡:0❛❜❡❧❡❝✐❞♦ 6✉❡ ♠T♥✐♠♦: :?♦ ❧♦❝❛❧♠❡♥0❡ ❞❡ ❝❧❛::❡ C1,1 ❡ ❡:0❛ 9 ❛ /❡❣✉❧❛/✐❞❛❞❡ C0✐♠❛ ♣❛/❛ :♦❧✉AF❡:✳ ❖

♣/♦❜❧❡♠❛ ❞❡ ❞✉❛: ❢❛:❡:✱ ✐✳❡✳✱ :❡♠ /❡:0/✐A?♦ :♦❜/❡ ♦ :✐♥❛❧ ❞♦ ❞❛❞♦ ❞❡ ❢/♦♥0❡✐/❛ ϕ✱ ❞❡:❛✜♦✉

❛ ❝♦♠✉♥✐❞❛❞❡ ♣♦/ ♠❛✐: ❞❡ 0/J: ❞9❝❛❞❛:✳ ❊:0✐♠❛0✐✈❛ C1,1 _{♣❛/❛ ♣/♦❜❧❡♠❛: ❞❡ ♦❜:0=❝✉❧♦} ❝♦♠ ❞✉❛: ❢❛:❡: ❢♦✐ ❡:0❛❜❡❧❡❝✐❞❛ ❡♠ ❬✺✺❪ ❝♦♠ ❛ ❛❥✉❞❛ ❞❛ ♣♦❞❡/♦:❛ ✉❛#❡ ❢C/♠✉❧❛ ❞❛ ♠♦♥♦0♦♥✐❝✐❞❛❞❡ ♦❜0✐❞❛ ❡♠ ❬✾❪✳

❖ ❝❛:♦ ✐♥❢❡/✐♦/✱ γ = 0✱ /❡❢❡/❡✲:❡ ❛ ♣/♦❜❧❡♠❛: ❞❡ ❥❡& ✢♦✇# ❡ ❝❛✈✐❞❛❞❡:✳ ❖ ♣/♦❜❧❡♠❛

❞❡ ❝❛✈✐0❛A?♦ ❞❡ ✉♠❛ ❢❛:❡ ❤♦♠♦❣J♥❡♦ ❡ ❧✐♥❡❛/ ❢♦✐ ❡:0✉❞❛❞♦ ❡♠ ❬✶❪✱ ♦♥❞❡ 9 ♣/♦✈❛❞♦ 6✉❡ ❛ /❡❣✉❧❛/✐❞❛❞❡ C0✐♠❛ ♣❛/❛ ❡:0❡ ♣/♦❜❧❡♠❛ ❛❧0❛♠❡♥0❡ :✐♥❣✉❧❛/ 9 C0,1✳ ❆ ✈❡/:?♦ ❝♦♠ ❞✉❛:

❢❛:❡: ❞❡:0❡ ♣/♦❜❧❡♠❛ ❛♣/❡:❡♥0❛ ♥♦✈❛: ❡ ♠❛✐♦/❡: ❞✐✜❝✉❧❞❛❞❡: ❡ ❛ /❡❣✉❧❛/✐❞❛❞❡ ▲✐♣:❝❤✐0③ ❞❡ ♠T♥✐♠♦: ❢♦✐ ♣/♦✈❛❞❛ ❡♠ ❬✷❪✱ ❝♦♠ ❛ ❛❥✉❞❛ ❞❛ /❡✈♦❧✉❝✐♦♥=/✐❛ ❢C/♠✉❧❛ ❞❡ ♠♦♥♦0♦♥✐❝✐❞❛❞❡ ❞❡ ❆❧0✲❈❛✛❛/❡❧❧✐✲❋/✐❡❞♠❛♥✱ ❞❡:❡♥✈♦❧✈✐❞❛ ♥❡:0❡ ❛/0✐❣♦ :❡♠✐♥❛❧✳ ▲✐♠✐0❛AF❡: ❞♦ ❣/❛❞✐❡♥0❡ ♣❛/❛ ♦ ♣/♦❜❧❡♠❛ ❞♦ 0✐♣♦ ❝❛✈✐0❛A?♦ ❤❡0❡/♦❣J♥❡♦ ❡ ❞❡ ❞✉❛: ❢❛:❡:✱ ✐✳❡✳✱ p =2✱ f∈ L∞

✱ ❡♠ ✭✶✳✸✮✱ ❢♦/❛♠ ❡:0❛❜❡❧❡❝✐❞❛: ♣♦/ ❈❛✛❛/❡❧❧✐✱ ❏❡/✐:♦♥ ❡ ❑❡♥✐❣ ❝♦♠ ❛ ❛❥✉❞❛ ❞❡ :✉❛ ♣♦❞❡/♦:❛ ❢C/♠✉❧❛ ❞❡ 6✉❛:❡✲♠♦♥♦0♦♥✐❝✐❞❛❞❡✱ ❬✾❪✳

❖ ♣/♦❜❧❡♠❛ ✐♥0❡/♠❡❞✐=/✐♦ 0 < γ < 1 0❛♠❜9♠ 0❡♠ /❡❝❡❜✐❞♦ ✉♠❛ ❣/❛♥❞❡ ❛0❡♥A?♦ ♥❛:

<❧0✐♠❛: ❞9❝❛❞❛:✳ ❖ ♣/♦❜❧❡♠❛ ❞❡ ❢/♦♥0❡✐/❛ ❧✐✈/❡ /❡❧❛❝✐♦♥❛❞♦ ♣♦❞❡ :❡/ ✉:❛❞♦✱ ♣♦/ ❡①❡♠♣❧♦✱ ♣❛/❛ ♠♦❞❡❧❛/ ❛ ❞❡♥:✐❞❛❞❡ ❞❡ ❝❡/0❛ ❡:♣9❝✐❡ 6✉T♠✐❝❛✱ ❡♠ /❡❛A?♦ ❝♦♠ ✉♠ ❝❛0❛❧✐:❛❞♦/ ♣♦/♦:♦ ♣❡❧❧❡0✳ ❆ ✈❡/:?♦ ❞♦ ♣/♦❜❧❡♠❛ ❞❡ ❢/♦♥0❡✐/❛ ❧✐✈/❡ ✭✶✳✸✮ ❤♦♠♦❣J♥❡♦✱ f ≡ 0✱ ❝♦♠ ✉♠❛ ❢❛:❡✱ ϕ ≥ 0 ❡ ❧✐♥❡❛/✱ p = 2✱ 9 ♦ 0❡♠❛ ❞❡ ✉♠ ♣/♦❣/❛♠❛ ❞❡ :✉❝❡::♦ ❞❡:❡♥✈♦❧✈✐❞♦ ♥♦: ❛♥♦:

✽✵ ♣♦/ H❤✐❧❧✐♣: ❡ ❆❧0 ❡ H❤✐❧❧✐♣:✱ ❬✺✷❪✱ ❬✺✶❪ ❡ ❬✸❪✱ ❡♥0/❡ ♦✉0/♦:✳ ❊♠ ❝♦♥0❡①0♦ :✐♠✐❧❛/✱ ❛ ❝♦♥0✐♥✉✐❞❛❞❡ ❍_❧❞❡/ ❞♦ ❣/❛❞✐❡♥0❡ ❞❡ ♠T♥✐♠♦: ❢♦✐ ♣/♦✈❛❞❛ ♣♦/ ●✐❛6✉✐♥0❛ ❡ ●✐✉:0✐ ❬✷✻❪✳ ▼❛✐: ✐♥✈❡:0✐❣❛AF❡: :♦❜/❡ ♦ ♣/♦❜❧❡♠❛ ❞❡ ❞✉❛: ❢❛:❡: ❡ ❧✐♥❡❛/ ❡♠ 6✉❡:0?♦ 0❛♠❜9♠ /❡6✉❡/ ❢C/♠✉❧❛: ❞❡ ♠♦♥♦0♦♥✐❝✐❞❛❞❡ ❡♠ :✉❛ ❛♥=❧✐:❡✳ ■:0♦ ❢♦✐ /❡❛❧✐③❛❞♦ ♣♦/ ❲❡✐:: ❬✻✺❪✳

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✻ ❞✐#♣♦♥'✈❡❧ ♣❛,❛ ✉♠ ♠'♥✐♠♦ ❞♦ ❢✉♥❝✐♦♥❛❧ Jγ✳ ❯♠❛ ✐♥❢❡,3♥❝✐❛ #✐♠♣❧❡# #♦❜,❡ ❛ ❡5✉❛67♦ ❞❡ ❊✉❧❡,✲▲❛❣,❛♥❣❡ ❢,❛❝❛ #❛<✐#❢❡✐<❛ ♣♦, ✉♠ ♠'♥✐♠♦✱ ✭✶✳✼✮ ❡ <❛♠❜B♠ ✭✶✳✽✮ ♣❛,❛ γ =0✱ ,❡✈❡❧❛

5✉❡∆pu❡①♣❧♦❞❡ ❛♦ ❧♦♥❣♦ ❞❛ ❢,♦♥<❡✐,❛ ❧✐✈,❡ ❞♦ ♣,♦❜❧❡♠❛✱ Fγ := ∂{uγ > 0}∪∂{uγ < 0}✳ E♦, ✐##♦✱ <♦,♥❛✲#❡ ❡##❡♥❝✐❛❧ ❡♥<❡♥❞❡, ❝♦♠♦ ❡#<❡ ❢❡♥F♠❡♥♦ ❛❢❡<❛ ❛# ♣,♦♣,✐❡❞❛❞❡# ❞❡ #✉❛✈✐❞❛❞❡ ❞♦ ♠'♥✐♠♦✳ ❙♦❜ <❛❧ ♣❡,#♣❡❝<✐✈❛✱ ❡ ❛❧❣✉♠❛ ❡①<❡♥#7♦✱ ❛ <❡♦,✐❛ ❞❡ ♣,♦❜❧❡♠❛# ❞❡ ❢,♦♥<❡✐,❛ ❧✐✈,❡ ❝♦♠ ❞✉❛# ❢❛#❡# ❣♦✈❡,♥❛❞♦# ♣♦, ♦♣❡,❛❞♦,❡# ❡❧'♣<✐❝♦#✱ ❞❡❣❡♥❡,❛❞♦# ❡ ♥7♦✲❧✐♥❡❛,❡# ❡#<H✱ ❛<B ❛❣♦,❛✱ ✐♥❛❝❡##'✈❡❧ ♥❛ ❧✐<❡,❛<✉,❛ ❛<✉❛❧✱ ♣,✐♥❝✐♣❛❧♠❡♥<❡ ♣❡❧❛ ✐♥❞✐#♣♦♥✐❜✐❧✐❞❛❞❡ ❞❡ ❢I,♠✉❧❛# ❞❡ ♠♦♥♦<♦♥✐❝✐❞❛❞❡ ♥❡#<❡ ❝♦♥<❡①<♦✳

◆♦ ❡#<✉❞♦ ❞❡ ♣,♦♣,✐❡❞❛❞❡# I<✐♠❛# ❞❡ #✉❛✈✐❞❛❞❡ ❞❡ ♠'♥✐♠♦# ❞♦ ❢✉♥❝✐♦♥❛❧ Jγ✱ ♠❛✐# ❞✐✜❝✉❧❞❛❞❡# #✉,❣❡♠ <❛♠❜B♠ ♣❡❧❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❛ <❡♦,✐❛ ❞❡ ,❡❣✉❧❛,✐❞❛❞❡ ♣❛,❛ ♦ ♦♣❡,❛❞♦, ❣♦✈❡,♥❛♥<❡ ∆p✳ ❘❡❝♦,❞❛♠♦# 5✉❡ ❢✉♥6M❡# p✲❤❛,♠F♥✐❝❛#✱ ✐✳❡✳✱ #♦❧✉6M❡# ♣❛,❛ ❡5✉❛67♦

❤♦♠♦❣3♥❡❛

∆ph =0 ❡♠ B1, #7♦ ❧♦❝❛❧♠❡♥<❡ ❞❡ ❝❧❛##❡ C1,αp ♣❛,❛ ✉♠ ❡①♣♦❡♥<❡ _{0 < α}

p < 1 5✉❡ ❞❡♣❡♥❞❡ #♦♠❡♥<❡ ❞❛ ❞✐♠❡♥#7♦ ❡ p✳ ❖ ✈❛❧♦, ♣,❡❝✐#♦ ❞❡αp B ❡♠ ❣❡,❛❧ ❞❡#❝♦♥❤❡❝✐❞♦✱ ✈❡❥❛ ❬✷✾❪ ♣❛,❛ ♦ ❝❛#♦ ♣❧❛♥❛,

n=2✳ ❊#<❡ ❢❛<♦ ✐♥❞✐❝❛ 5✉❡ ❡#<✐♠❛<✐✈❛# ✐♥<❡,✐♦,❡# ❞✐#♣♦♥'✈❡✐# ♣❛,❛ ❢✉♥6M❡# p✲❤❛,♠F♥✐❝❛#✱

5✉❡ #7♦ ✐♥❢❡,✐♦,❡# U 5✉❛❞,H<✐❝❛✱ C1,1_{✱ ❝♦♠♣❡<✐,7♦ ❝♦♠ ♦ ❝,❡#❝✐♠❡♥<❡ I<✐♠♦ ❛♦ ❧♦♥❣♦ ❞❛} ❢,♦♥<❡✐,❛ ❧✐✈,❡Fγ✳ ❆ <❡♦,✐❛ ❞❡ ,❡❣✉❧❛,✐❞❛❞❡ ♣❛,❛ ❡5✉❛6M❡# ❤❡<❡,♦❣3♥❡❛# ∆pξ=f(X)B ❛✐♥❞❛ ♠❛✐# ❡♥✈♦❧✈✐❞❛ ❡✱ ❛<B ♦ ♥♦##♦ ❝♦♥❤❡❝✐♠❡♥<♦✱ ❛ ❝♦♠♣,❡❡♥#7♦ ❞❡#<❛ ❝❧❛##❡ ❞❡ ♣,♦❜❧❡♠❛# ❛✐♥❞❛ ♥7♦ ❡#<H <♦<❛❧♠❡♥<❡ ❝♦♠♣❧❡<❛✳

❆ ❝♦♥<,✐❜✉✐67♦ ❢✉♥❞❛♠❡♥<❛❧ ❞❡#<❡ <,❛❜❛❧❤♦ ❡#<H ❡♠ ✉♠❛ ❞❡#❝,✐67♦ ❝♦♠♣❧❡<❛ ❞❛ ❞❡#❝♦♥❤❡❝✐❞❛ <❡♦,✐❛ ❞❡ ,❡❣✉❧❛,✐❞❛❞❡ I<✐♠❛ ♣❛,❛ ♠'♥✐♠♦# ❞❡ Jγ✱ ♦ 5✉❡ ❛❜,❡ ❛ ♣♦##✐❜✐❧✐❞❛❞❡ ❞❡ ✐♥✈❡#<✐❣❛, ❡#<❛ ❝❧❛##❡ ❞❡ ♣,♦❜❧❡♠❛# ❡♠ <♦❞❛ #✉❛ ❣❡♥❡,❛❧✐❞❛❞❡✳

❉♦ ♣♦♥<♦ ❞❡ ✈✐#<❛ ♠❛<❡♠H<✐❝♦✱ ♦ ❡①♣♦❡♥<❡ γ 5✉❡ ❛♣❛,❡❝❡ ❡♠ ✭✶✳✸✮ #❡,H ❝♦♠♣,❡❡♥❞✐❞♦

❝♦♠♦ ♦ ♣❛,Y♠❡<,♦ 5✉❡ ♠❡❞❡ ❛ #✐♥❣✉❧❛,✐❞❛❞❡ ❞♦ <❡,♠♦ ❞❡ ❛❜#♦,67♦ ❞❛ ❡5✉❛67♦ ,❡❧❛❝✐♦♥❛❞❛✳ E❛,❛ ❢✉♥❝✐♦♥❛✐# ♥7♦✲❞✐❢❡,❡♥❝✐H✈❡✐#✱ ♠❛# ❝♦♥<'♥✉♦#✱ Jγ ❝♦♠ 0 < γ ≤ 1 ❢♦✐ ❝♦♥❥❡❝<✉,❛❞♦ 5✉❡ ♦ ❣,❛❞✐❡♥<❡ ❞❡ ✉♠ ♠'♥✐♠♦ B ❧♦❝❛❧♠❡♥<❡ ❍[❧❞❡, ❝♦♥<'♥✉♦✱ ♠❡#♠♦ ❛<,❛✈B# ❞❛ ❢,♦♥<❡✐,❛ ❧✐✈,❡ #✐♥❣✉❧❛, Fγ✳ ❖ ♣,✐♠❡✐,♦ ,❡#✉❧<❛❞♦ 5✉❡ ❛♣,❡#❡♥<❛♠♦# ♥❡#<❡ <,❛❜❛❧❤♦ ❞H ✉♠❛ ,❡#♣♦#<❛ ❛✜,♠❛<✐✈❛ ♣❛,❛ <❛❧ 5✉❡#<7♦✳ ❆❞❡♠❛✐#✱ ❢♦,♥❡❝❡♠♦# ❛ <❡♦,✐❛ ❞❡ ,❡❣✉❧❛,✐❞❛❞❡ ✐♥<❡,✐♦, C1,α _❡ ❛!!✐♥$♦$✐❝❛♠❡♥$❡ I<✐♠❛ ❞✐#♣♦♥'✈❡❧ ♣❛,❛ ♠'♥✐♠♦# ❞❡ <❛✐# ❢✉♥❝✐♦♥❛✐#✳

❚❡♦#❡♠❛ ✶ ✭❘❡❣✉❧❛,✐❞❛❞❡ C1,α_{✮✳ ❙❡❥❛} _u _{✉♠ ♠,♥✐♠♦ ❞♦ ♣/♦❜❧❡♠❛ ✭✶✳✸✮✳ ❆!!✉♠❛ 4✉❡}

0 < γ ≤1 ❡ f∈Lq₍_Ω₎_{✱ ♣❛/❛ ❛❧❣✉♠} _{q > n}_{✳ ❊♥$8♦} _u_∈_C1,α ❧♦❝✱ ♣❛/❛

α:= min

α−p,

γ p−γ,

(q−n) (p−1)q

. ✭✶✳✾✮

(19)

✼

Ω′_✱ _n_✱ _p_✱ _q_✱ _k_ϕ_k

L∞₍_Ω₎✱ kfk_Lq₍_Ω₎✱ λ₊✱ λ₋✱ γ ❡ (α_p−α)✱ "❛❧ %✉❡

(20)

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STACK:

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