Graduate School of Fundamental Science and Engineering Waseda University

Graduate School of Fundamental Science and Engineering Waseda University

༤

༤㻌 ኈ㻌 ㄽ㻌 ᩥ㻌 ᴫ㻌 せ

Doctoral Thesis Synopsis

ㄽ ᩥ 㢟 ┠ T h e s i s T h e m e

Verified numerical computation for elliptic partial differential equations and related

problems

ᴃ෇ᆺ೫ᚤศ᪉⛬ᘧ࡟ᑐࡍࡿ⢭ᗘಖド௜ࡁ

ᩘ್ィ⟬࡜㛵㐃ࡍࡿၥ㢟

⏦ ㄳ ⪅ (Applicant name)

Kazuaki TANAKA ⏣୰ ୍ᡂ

Department of Pure and Applied Mathematics Research on Numerical Analysis

December, 2016

ຊ࿦จͰ͸ҎԼͷପԁܕσΟϦΫϨڥք஋໰୊(1)ʹର͢Δਫ਼౓อূ෇͖਺஋ܭ

ࢉ ๏ Λ ߟ ͑ Δ ɽ

−Δu(x) =f(u(x)), x∈Ω,

u(x) = 0, x∈∂Ω. (1)

͜͜ͰɼΩ^͸R²্ͷ༗քͳଟ֯ܗྖҬɼଈͪ༗քͳଟ֯ܗঢ়ͷ։ू߹ͱ͢Δɽ∂Ω

͸ͦͷڥքΛද͢ɽ·ͨɼΔ^{͸ϥϓϥε࡞༻ૉɼ}f :R→R͸༩͑ΒΕͨඇઢܗࣸ૾

ͱ͢Δɽຊ࿦จʹ͓͍ͯ໰୊(1)^ʹର͢Δ“^ਫ਼౓อূ”^ͱ͸ɼ(1)^ͷۙࣅղuˆ^Λ਺஋త ʹ ٻ Ί ɼͦ Ε Λ த ৺ ͱ ͢ Δ ໌ ࣔ ͞ Ε ͨ ൒ ܘr^{ͷ ด ٿ}B(ˆu, r)಺ ʹ ਅ ͷ ղ ͕ ଘ ࡏ ͢ Δ

͜ͱΛূ໌͢Δͱ͍͏͜ͱΛࢦ͢ɽͨͩ͠ɼຊ࿦จͰ͸H₀^1ϊϧϜ(^ଈͪ∇ · _L²_(Ω)) ͱL^∞ϊ ϧ Ϝ ͷ ҙ ຯ Ͱ ͷ ਫ਼ ౓ อ ূ Λ ߟ ͑ Δ ɽຊ ࿦ จ ͷ ର ৅ ͸ Ҏ Լ ͷ ໰ ୊(2)^{ͷ ղ ʹ} ର ͢ Δ ਫ਼ ౓ อ ূ ෇ ͖ ਺ ஋ ܭ ࢉ Λ ؚ Ή ɽ

⎧⎪

⎨

⎪⎩

−Δu(x) =f(u(x)), x∈Ω, u(x)>0, x∈Ω, u(x) = 0, x∈∂Ω.

(2)

ଈͪɼඞཁͰ͋Δ৔߹͸(1)ͷਫ਼౓อূ͞Εͨղʹରͯ͠ɼͦͷਖ਼஋ੑ΋ಉ࣌ʹอ

ূ ͢ Δ ͜ ͱ Ͱ(2)ͷ ղ Λ ਫ਼ ౓ อ ূ ͢ Δ ɽ໰ ୊(1)^{ͷ ղ}u^{͕ ໰ ୊}(2)ͷ ղ Ͱ ΋ ͋ Δ ͱ

͖ ɼu^Λ(1)ͷ ਖ਼ ஋ ղ ͱ ͍ ͏ɽ໰ ୊ ͷ Մ ղ ੑ ΍ ਫ਼ ౓ อ ূ ෇ ͖ ਺ ஋ ܭ ࢉ Λ ߦ ͏ ೉ ͠ ͞

͸ඇઢܗࣸ૾fʹେ͖͘ґଘ͢Δ͕ɼຊ࿦จͰ͸୅දతͳྫͱͯ͠ɼf(t) =|t|^p−1t (p >1)^{͓ Α ͼ}f(t) =ε⁻²(t−t³)Λ औ Γ ѻ ͏ɽ͜ ͜ Ͱ ɼε͸ ಛ ҟ ઁ ಈ ͱ ݺ ͹ Ε Δ ݱ ৅ ʹ ؔ ܎ ͢ Δ ਖ਼ ͷ খ ͞ ͍ ύ ϥ ϝ ʔ λ ʔ Ͱ ͋ Δ ɽຊ ࿦ จ ͷ ͍ ͘ ͭ ͔ ͷ Օ ॴ ʹ ͯΩ^{ͷ ತ}

ੑ ͕ Ճ ͑ ͯ ཁ ٻ ͞ Ε Δ ɽ͜ Ε ͸ ਫ਼ ౓ อ ূ ͞ Ε ͨ(1)^{ͷ ऑ ղ}u ∈H₀¹(Ω)^ͷH^{2ਖ਼ ଇ ੑ}

͕L^∞ϊ ϧ Ϝ ͷ ҙ ຯ Ͱ ͷ ਫ਼ ౓ อ ূ Λ ߦ ͏ ࡍ ʹ ඞ ཁ ͱ ͳ Δ ͨ Ί Ͱ ͋ Δ ɽ

ࠓ ೔ ɼ਺ ஋ ܭ ࢉ ͸ ੜ ෺ ֶ ΍ ෺ ཧ ֶ ౳ ʹ ༝ དྷ ͢ Δ ༷ʑͳ ݱ ৅ Λ ཧ ղ ͢ Δ ͨ Ί ʹ Պ

ֶ ͓ Α ͼ ޻ ֶ ͷ ޿ ͍ ෼ ໺ Ͱ ॏ ཁ ͳ ໾ ׂ Λ ୲ͬͯ ͍ Δ ɽ͠ ͔ ͠ ै དྷ ͷ ਺ ஋ ܭ ࢉ ͸ Ұ ൠʹؙΊޡࠩ΍ଧ੾Γޡࠩɼ཭ࢄԽޡࠩΛ͸͡Ίͱ͢Δ༷ʑͳޡࠩΛ൐͍ɼͦΕΒ

͕ ࠷ ऴ త ͳ ਺ ஋ ݁ Ռ ʹ க ໋ త ͳ Ө ڹ Λ ٴ ΅ ͢ ͜ ͱ ͕ ͋ Δ ͨ Ί ɼ͜ ͷ ͜ ͱ ͸ ༧ ͯ Α Γ ໰ ୊ ࢹ ͞ Ε ͯ ͖ ͨ ɽҰ ํ Ͱ ਺ ஋ ܭ ࢉ ͷ ա ఔ Ͱ ൃ ੜ ͢ Δ શ ͯ ͷ ޡ ࠩ Λ ߟ ྀ ͠ ɼఆ

ྔ త ʹ ޡ ࠩ ্ ݶ Λ อ ূ ͢ Δ ਺ ஋ ܭ ࢉ ͷ ͜ ͱ Λ“ਫ਼ ౓ อ ূ ෇ ͖ ਺ ஋ ܭ ࢉ”^{ͱ ݺ Ϳ ɽ͜}

͜ Ͱ“^{ਫ਼ ౓ อ ূ}”ͱ ͍ ͏ ݴ ༿ ʹ ͸ ޡ ࠩ ͷ ೺ Ѳ Ҏ ֎ ʹ ɼର ৅ ͱ ͢ Δ ໰ ୊ ͷ ղ ͷ ଘ ࡏ ͷ อ ূ Λ ಉ ࣌ ʹ ߦ ͏ ͱ ͍ ͏ ҙ ຯ Λ ؚ Ή ɽͦ ͷ ͨ Ί ɼ਺ ֶ త ཱ ৔ ͔ Β ਺ ஋ త ݕ ূ ๏ ΍ ܭࢉػԉ༻(^ଘࡏ)ূ໌ͱݺ͹ΕΔ͜ͱ΋͋Δɽઌߦݚڀͱͯ͠͸ʰେੴਐҰ,^ਫ਼౓

อ ূ ෇ ͖ ਺ ஋ ܭ ࢉ ,^{ί ϩ φ ࣾ} , 2000ʱ΍ʰ த ඌ ॆ ޺ ɼ౉ ෦ ળ ོ ,࣮ ྫ Ͱ ֶ Ϳ ਫ਼ ౓ อ

ূ ෇ ͖ ਺ ஋ ܭ ࢉ ཧ ࿦ ͱ ࣮ ૷ , α Π Τ ϯ ε ࣾ , 2011ʱ౳ ʹ · ͱ Ί ͯ ͋ Δ ख ๏ ͕ ༗ ໊ Ͱ ͋ Γɼਫ਼ ౓ อ ূ ෇ ͖ ਺ ஋ ܭ ࢉ ͸ ै དྷ ͷ ਺ ஋ ܭ ࢉ ʹ ͸ ͳ ͍ ಛ ௃ Λ ࣋ ͭ ͜ ͱ ͔ Β ۙ

೥ ੈ ք త ʹ ஫ ໨ Λ ू Ί ͯ ͍ Δ ٕ ज़ Ͱ ͋ Δ ɽ

ಛ ʹ զʑͷ ڵ ຯ ͸ ପ ԁ ܕ ڥ ք ஋ ໰ ୊(1)^{͓ Α ͼ}(2)ʹ ର ͢ Δ ਫ਼ ౓ อ ূ ෇ ͖ ਺ ஋ ܭ

ࢉͰ͋Δɽ(1)^͓Αͼ(2)͸ੜ෺ֶ΍෺ཧֶ౳ʹ༝དྷ͢Δ༷ʑͳϞσϧʹىҼ͢Δॏ

1

ཁ ͳ ภ ඍ ෼ ํ ఔ ࣜ Ͱ ͋ Δ ͨ Ί ɼ͜ Ε · Ͱ ղ ੳ త ɼ਺ ஋ త ྆ ํ ͷ ଆ ໘ ͔ Β ޿ ͘ ݚ ڀ

͞Ε͖ͯͨɽಛʹຊ࿦จͰର৅ͱ͢Δf(t) =|t|^p−1t(p >1)^͓Αͼf(t) =ε⁻²(t−t³)

(ε >0)^{ͷͲͪΒͷ৔߹΋}(2)ͷղͷଘࡏੑɾ།Ұੑʹؔ͢Δ݁Ռ͕ภඍ෼ํఔࣜ࿦

ʹΑΓ஌ΒΕ͍ͯΔɽf(t) =|t|^p−1t(p >1)^ͷͱ͖͸ɼΩ͕׈Β͔Ͱ͔֤ͭ࣠ʹର͠

ͯ ର শ Ͱ ͋ Ε ͹(2)͸ ། Ұ ղ Λ ࣋ ͭ ɽ· ͨ ɼྖ Ҭ ͕ ׈ Β ͔ Ͱ ͳ ͍ ৔ ߹ ΋Ω = (0,1)² ͷ ౳ ͷ ղ ͷ े ෼ ͳ ׈ Β ͔ ͞ ͕ อ ূ ͞ Ε Δ ର শ ྖ Ҭ Ͱ ͋ Ε ͹ ಉ ༷ ʹ ਖ਼ ஋ ղ ͷ ། Ұ ੑ

͕อূ͞ΕΔɽf(t) =ε⁻²(t−t³) (ε >0)^ͷͱ͖͸ɼλ₁^Λ−Δ^{ͷσΟϦΫϨ৚݅Λ՝}

ͨ͠Ω্Ͱͷऑ͍ҙຯͰͷ࠷খݻ༗஋ͱ͢Δͱ͖ɼε⁻²> λ₁^Ͱ͋Ε͹(2)^͸།Ұղ Λ ࣋ ͭ ͱ ͍ ͏ ͜ ͱ ͕ ೚ ҙ ͷ ༗ ք ྖ Ҭ ʹ ର ͠ ͯ ಘ Β Ε Δ ɽ͜ ͷ Α ͏ ͳ ݁ Ռ ͕ ஌ Β Ε

ͯ ͍ Δ ʹ ΋ ؔ Θ Β ͣɼղ ͷ ఆ ྔ త ͳ ৘ ใ(֤ ఺ Ͱ ͷ ஋ ΍ ࠷ େ ஋ ɼ࠷ খ ஋ ౳)^{΍ ͦ ͜}

͔ Β ؍ ଌ ͞ Ε Δ ղ ͷ ܗ ঢ় ౳ ͸ ɼݱ ঢ় ͷ ղ ੳ త ख ๏ ͷ Έ ͔ Β Ͱ ͸ ໌ Β ͔ ʹ ͞ Ε ͯ ͍ ͳ ͍ ɽਫ਼ ౓ อ ূ ෇ ͖ ਺ ஋ ܭ ࢉ ͸ ɼਅ ͷ ղ ͷ ఆ ྔ త ͳ ৘ ใ Λ ͦ ͷ ਺ ஋ త ʹ ಘ ͨ ۙ ࣅ ղ ͱ ͷ ࠩ Λ ද ͢ ෆ ౳ ࣜ ͷ ܗ Ͱ ಘ Δ ͜ ͱ ͕ Ͱ ͖ Δ ͱ ͍ ͏ ఺ Ͱ ༗ ༻ Ͱ ͋ Δ ͱ ݴ ͑ Δ ɽ

Ҏ Լ ɼຊ ࿦ จ ͷ ֤ ষ ͷ ֓ ཁ Λ ड़ ΂ Δ ɽ

1ষͰ͸ຊ࿦จͷର৅ͱ͢Δ໰୊΍Ϟνϕʔγϣϯɼഎܠɼઌߦݚڀ౳ʹ͍ͭͯ

ه ड़ ͢ Δ ɽ

2ষ Ͱ ͸ · ͣ ຊ ࿦ จ Ͱ ࢖ ༻ ͢ Δ ه ߸ ͷ ४ උ Λ ߦ ͏ɽ࣍ ʹ ໰ ୊(1)^{ʹ ର ͢ Δ}H₀^1ϊ ϧϜͷҙຯͰͷਫ਼౓อূ෇͖਺஋ܭࢉͷͨΊʹ༻͍ΔχϡʔτϯɾΧϯτϩϏον ͷఆཧ͓Αͼͦͷվྑ͞ΕͨఆཧΛ঺հ͢Δɽ͜ΕΒͷఆཧ͸(1)^ͷۙࣅղuˆ^͕͋

Δछͷ“^ྑ͍”৚ ݅Λ ຬ ͨͤ ͹ ͦ ͷ ۙ͘ ʹਅ ͷղ ͕ଘ ࡏ͢ Δͱ ͍͏ ͜ͱ Λओ ு͢ Δ

΋ͷͰ͋Δɽ·ͨɼͦ͜ʹߋʹ৚݅ΛՃ͑Δ͜ͱʹΑΓ(2)^ʹର͢Δ(^ଈͪ(1)^ͷਖ਼

஋ղʹର͢Δ)ਫ਼౓อূ΋ՄೳʹͳΔͱ͍͏͜ͱ΋ड़΂ΔɽྖҬ͕ತͰ͋Δ৔߹͸

໰ ୊(1)^{ͷ ऑ ղ ͷ}H²ਖ਼ ଇ ੑ ͕ อ ূ ͞ Ε Δ ͨ Ί ɼH²(Ω)^{͔ Β}L^∞(Ω)΁ ͷ ຒ Ί ࠐ Έ ఆ

਺Λ۩ମతʹධՁ͢Δ͜ͱʹΑΓɼ্هख๏ʹΑΓಘΒΕͨH₀^{1ϊϧϜͷҙຯͰͷ} ղ ͷ แ ؚ ͔ Β ɼL^∞(Ω)ͷ ҙ ຯ Ͱ ͷ ղ ͷ แ ؚ ͕ ಘ Β Ε Δ ͜ ͱ ΋ ड़ ΂ Δ ɽຊ ষ Ͱ ঺ հ

͢ Δ ఆ ཧ ͷ ཁ ٻ ͢ Δ ۙ ࣅ ղuˆͷ ຬ ͨ ͢ ΂ ͖ ৚ ݅ Λ νΣοΫ ͢ Δ ͨ Ί ʹ ͸ ɼ໰ ୊(^ྖ ҬΩ΍ ඇ ઢ ܗ ࣸ ૾f)^΍uˆʹ ґ ଘ ͢ Δ ͍ ͘ ͭ ͔ ͷ ఆ ਺ Λ ۩ ମ త ʹ ධ Ձ ͢ Δ ඞ ཁ ͕ ͋ Δ ɽྫ ͑ ͹ ɼ໰ ୊ Λ ه ड़ ͢ Δ ࡞ ༻ ૉ ͷuˆʹ ͓ ͚ Δ ઢ ܗ Խ ࡞ ༻ ૉ ͷ ٯ ࡞ ༻ ૉ ϊ ϧ Ϝ

΍ɼຒΊࠐΈH₀¹(Ω)→L^p(Ω)^{ͷෆ౳ࣜʹݱΕΔఆ਺}C_p(Ω)^౳Ͱ͋Δɽ2^ষͷ࠷ޙͰ

͸C_p(Ω)ͷ ্ ͔ Β ͷ ૈ ͍ ධ Ձ Λ ಘ Δ ͨ Ί ͷ ݹ య త ͳ ख ๏ Ͱ ͋ Δ λ ϥ ϯ ςΟͷ ࠷ ྑ ఆ ਺ ͱ ϔ ϧ μ ʔ ͷ ෆ ౳ ࣜ Λ ༻ ͍ ͨ ධ Ձ ํ ๏ ʹ ͭ ͍ ͯ ΋ ݴ ٴ ͢ Δ ɽ

3ষ Ͱ ͸ ɼର ৅ ͱ ͢ Δ ໰ ୊(1)Λ ऑ ܗ ࣜ Խ ͠ ࡞ ༻ ૉF : H₀¹(Ω) → H⁻¹(Ω) (u →

−Δu−f(u))^Λ༻͍ͯF(u) = 0ͱॻ͍ͨͱ͖ʹɼͦͷۙࣅղuˆ^ʹ͓͚ΔF^ͷϑϨο γΣඍ෼F_uˆ^{ͷٯ࡞༻ૉϊϧϜ}F_uˆ⁻¹

B(H⁻¹(Ω),H₀¹(Ω))ͷධՁ๏ʹ͍ͭͯड़΂Δɽ͜͜

ͰH⁻¹(Ω)^͸H₀¹(Ω)ͷ ڞ ໾ ۭ ؒ Λ ද ͠ ɼ௨ ৗ ͷsupϊ ϧ Ϝ Λ ಋ ೖ ͠ ͨ ΋ ͷ ͱ ͢ Δ ɽ

͜ ͷ ٯ ࡞ ༻ ૉ ϊ ϧ Ϝ ͸uˆʹ Α Γ ઁ ಈ ͞ Ε ͨ ͋ Δ ପ ԁ ܕ ࣗ ݾ ڞ ໾ ࡞ ༻ ૉ ͷ ࠷ খ ݻ ༗

஋ ͔ Β ܭ ࢉ ͢ Δ ͜ ͱ ͕ Ͱ ͖ ɼͦ ͷ ࠷ খ ݻ ༗ ஋ Λ ཱུɾେ ੴ ͷ ख ๏ Ͱ ্ Լ ͔ Β ۩ ମ త

ʹ ධ Ձ ͢ Δ ͜ ͱ Ͱ ॴ ๬ ͷ ϊ ϧ Ϝ ධ Ձ Λ ࣮ ݱ ͢ Δ ɽཱུɾେ ੴ ͷ ख ๏ ͸H₀¹(Ω)^{ͷ ͋ Δ}

༗ݶ࣍ݩ෦෼ۭؒVN্Ͱۙࣅ͞Εͨݻ༗஋ͱɼݩͷແݶ࣍ݩۭؒH₀¹(Ω)^Ͱܭࢉ͞

Ε ͨ ݻ ༗ ஋ ͷ ࠩ Λ ධ Ձ ͢ Δ ɽ͜ ͜ ͰN ^͸V_N ^ͷH₀¹(Ω)ʹ ର ͢ Δ ۙ ࣅ ౓ Λ ද ͢ ਖ਼ ͷ ύ ϥ ϝ ʔ λ Ͱ ͋ Δ ɽྫ ͑ ͹ ຊ ࿦ จ Ͱ ͸{φ_i}^∞_i=1 ^ΛH₀¹(Ω)Λ ு Δ ج ఈ ͱ ͠ ͨ ͱ ͖ ʹ ɼ V_N =span{φ_i :i= 1,2,· · ·, N}^ͷΑ͏ʹV_N^{Λબͼɼ͜ͷͱ͖}N^͸V_N^{ΛுΔجఈؔ}

਺ ͷ ਺ Λ ද ͢ ͜ ͱ ʹ ͳ Δ ɽཱུɾେ ੴ ͷ ख ๏ Λ ༻ ͍ ͯ ର ৅ ͱ ͢ Δ ݻ ༗ ஋ Λ ධ Ձ ͢ Δ ʹ͸V_N ^ͷH₀¹(Ω)ʹର͢Δۙࣅ౓Λ൓ө͢Δิ׬ޡࠩఆ਺C_N ^{Λ۩ମతʹධՁ͢Δ} ඞཁ͕͋Δɽଈͪɼh∈L²(Ω)ʹରͯ͠ɼͦΕʹରԠ͢ΔϙΞιϯํఔࣜ−Δu=h ͷ σΟϦ Ϋ Ϩ ڥ ք ஋ ໰ ୊ ͷ ऑ ղ Λu_hͱ ॻ ͍ ͨ ͱ ͖ ʹ

u_h−P_Nu_hV ≤C_Nh_L2(Ω) (3) Λ ೚ ҙ ͷh ∈ L²(Ω)ʹ ର ͠ ͯ ຬ ͨ ͢ ਖ਼ ఆ ਺CN Ͱ ͋ Δ ɽ͜ ͜ Ͱ ɼPN ͸H₀¹(Ω)^{͔ Β} VN ΁ ͷ ௚ ަ ࣹ Ө Ͱ ͋ Γɼ(PNu−u, vN)_H0¹_(Ω) = 0 for all u ∈ H₀¹(Ω) andvN ∈ VN Ͱ ఆ ٛ ͞ Ε Δ ɽ3ষ ͷ ࠷ ޙ Ͱ ͸VN Λ ϧ δϟϯ υ ϧ ଟ ߲ ࣜ ͔ Β ߏ ੒ ͞ Ε Δ ج ఈ Ͱ ு Β Ε ͨ ۭ ؒ ͱ ͠ ͨ ͱ ͖ ͷCNͷ ධ Ձ ๏ Λ ड़ ΂ Δ ɽ

4ষ Ͱ ͸ ؔ ਺ ۭ ؒ ؒ ͷ ຒ Ί ࠐ ΈH₀¹(Ω)→ L^p(Ω)^{ͷ ࠷ ྑ ఆ ਺}C_p(Ω)ͷ ධ Ձ ๏ ʹ ͭ

͍ ͯ ड़ ΂ Δ ɽΑ Γ ਖ਼ ֬ ʹ ͸C_p(Ω)^͸

u_Lp(Ω)≤Cp(Ω)u_H0¹_(Ω) for allu∈H₀¹(Ω). (4) Λ ຬ ͨ ͢ ਖ਼ ఆ ਺ ͱ ͠ ͯ ఆ ٛ ͞ Ε Δ ɽ͜ ͷ ࠷ ྑ ஋(^{ଈ ͪ ۃ ஋})Λ ୡ ੒ ͞ ͤ Δ ؔ ਺u^͸ ପ ԁ ܕ ڥ ք ஋ ໰ ୊

⎧⎪

⎨

⎪⎩

−Δu=u^p−1 in Ω, u >0 in Ω,

u= 0 on ∂Ω

(5)

Λ ຬ ͨ ͢ ͜ ͱ Λ ࣔ ͢ɽߋ ʹ ໰ ୊(5)ͷ ղ ͷ ། Ұ ੑ ʹ ͭ ͍ ͯΩ = (0,1)²ͷ ৔ ߹ ʹ ٞ ࿦

͠ ɼ2ষ Ͱ ड़ ΂ ͨ ํ ๏ Ͱ ͦ ͷ ղ ͷ ਫ਼ ౓ อ ূ Λ ߦ ͏ ͜ ͱ ʹ Α ΓɼC_p(Ω)ͷ ࠷ ྑ ஋ Λ λ Π τ ʹ ධ Ձ ͢ Δ ɽಛ ʹ ຊ ষ Ͱ ͸p = 3,4,5,6,7ͷ ৔ ߹ ͷ ݁ Ռ Λ ༩ ͑ ɼC_p(Ω)^{ͷ ࠷ ྑ}

஋ Λ12^ʙ13ܻ ͷ ਫ਼ ౓ Ͱ ධ Ձ ͠ ͨ ݁ Ռ Λ · ͱ Ί Δ ɽ

5^{ষ Ͱ ͸}f(t) =ε⁻²(t−t³)ͷ ৔ ߹ ɼଈ ͪ Ξ Ϩ ϯɾΧ ʔ ϯ ํ ఔ ࣜ ͷ ఆ ৗ ໰ ୊ −Δu=ε⁻²(u−u³) in Ω,

u= 0 on ∂Ω (6)

ʹର͢Δਫ਼౓อূ෇͖਺஋ܭࢉΛߟ͑Δɽ͜ͷ໰୊ͷղ͸ε >0^{͕খ͘͞ͳΔʹͭ}

Ε ͯ ಛ ҟ ઁ ಈ ͱ ͍ ͏ ݱ ৅ Λ ى ͜ ͢ɽ· ͨ ɼෳ ࡶ ͳ ղ ͷ ෼ ذ Λ Ҿ ͖ ى ͜ ͢ ͜ ͱ Ͱ ΋

஌ΒΕΔɽ͜ͷ͜ͱ͔ΒҰൠʹε͕খ͍͞΄Ͳղͷਫ਼౓อূ͸೉͘͠ͳΔɽ5^ষͰ

͸εΛ ม Խ ͞ ͤ ͨ ৔ ߹ ʹ ਫ਼ ౓ อ ূ ʹ ඞ ཁ ͳ ֤ छ ఆ ਺ ΍ ղ ͷ ܗ ͕ Ͳ ͷ Α ͏ ʹ ม Խ ͢ Δ͔Λ۩ମతͳྫͱڞʹ঺հ͢Δɽ·ͨɼ2ষͰड़΂ͨํ๏ΛԠ༻͠ɼͦͷਖ਼஋ղ ͷ ਫ਼ ౓ อ ূ Λ ߦͬͨ ྫ ΋ ซ ͤ ͯ ঺ հ ͢ Δ ɽ

6ষ Ͱ ͸ ຊ ࿦ จ Ͱ ड़ ΂ ͨ ख ๏ɾ݁ Ռ Λ ૯ ׅ ͠ ɼࠓ ޙ ͷ ల ๬ Λ ड़ ΂ Δ ɽ

3

㹌㹭

᪩

᪩✄⏣኱Ꮫ ༤ኈ㸦ᕤᏛ㸧 Ꮫ఩⏦ㄳ ◊✲ᴗ⦼᭩

Ặྡ ⏣୰ ୍ᡂ ༳

㸦 ᖺ ᭶ ⌧ᅾ㸧

✀㢮ู 㢟ྡࠊ Ⓨ⾲࣭Ⓨ⾜ᥖ㍕ㄅྡࠊ Ⓨ⾲࣭Ⓨ⾜ᖺ᭶ࠊ 㐃ྡ⪅㸦⏦ㄳ⪅ྵࡴ㸧

ㄽᩥ

ㅮ₇

[1] Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, Shin'ichi Oishi: Numerical validation of blow-up solutions for ODEs, to appear in Journal of Computational and Applied Mathematics.

[2] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, and Shin'ichi Oishi: Sharp numerical inclusion of the best constant for embedding on bounded convex domain, Journal of Computational and Applied Mathematics, 311, 306–313 (2017), to appear.

Electronically published in doi.org/10.1016/j.cam.2016.07.021.

[3] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, and Shin'ichi Oishi: Estimation of the Sobolev embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and Applications, 1, 1-23 (2015).

[4] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, and Shin'ichi Oishi: Numerical verification of positiveness for solutions to semilinear elliptic problems, JSIAM Letters 7, 73-76 (2015).

[5] Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu, and Shin'ichi Oishi: Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation, Japan Journal of Industrial and Applied Mathematics, 31, 665-679 (2014).

[6] ⏣୰୍ᡂ, 㛵᰿᫭ኴ, ኱▼㐍୍㸸ᴃ෇ᆺᚤศ᪉⛬ᘧࡢṇ್ゎ࡟ᑐࡍࡿ⢭ᗘಖド௜ࡁᩘ

್ ィ ⟬ ἲ 㸦Verified numerical computation method for positive solutions to elliptic differential equations㸧, RIMS◊✲㞟఍ࠕ⌧㇟ゎ᫂࡟ྥࡅࡓᩘ್ゎᯒᏛࡢ᪂ᒎ㛤IIࠖ, 2016ᖺ10᭶19᪥㹼10᭶21᪥㸬

[7] 㛵᰿᫭ኴ, ⏣୰୍ᡂ, ኱▼㐍୍㸸࠶ࡿ↓㝈ḟඖᅛ᭷್ࢆ⏝࠸ࡓᴃ෇ᆺ೫ᚤศ᪉⛬ᘧࡢ ゎࡢᏑᅾᛶ࡟ᑐࡍࡿィ⟬ᶵ᥼⏝ド᫂ἲ 㸦Computer-assisted proof for existence of solutions to PDEs using an infinite eigenvalue㸧, RIMS◊✲㞟఍ࠕ⌧㇟ゎ᫂࡟ྥࡅࡓᩘ

್ゎᯒᏛࡢ᪂ᒎ㛤IIࠖ, 2016ᖺ10᭶19᪥㹼10᭶21᪥㸬

[8] 㛵᰿᫭ኴ, ⏣୰୍ᡂ, ኱▼㐍୍㸸᭷⏺࡞ฝ㡿ᇦ࡟࠾ࡅࡿ㐃❧ᴃ෇ᆺ೫ᚤศ᪉⛬ᘧࡢゎ ࡢィ⟬ᶵ᥼⏝Ꮡᅾド᫂ἲ㸦Computer assisted existence proof of solutions to system of partial differential equations with bounded convex polygonal domains㸧, The Twenty-Eighth RAMP Symposium, 2016ᖺ10᭶13᪥㹼10᭶14᪥㸬

[9] Kazuaki Tanaka, Kouta Sekine, Shin'ichi Oishi: On verified numerical computation for positive solutions to elliptic boundary value problems, Computer Arithmetic and Validated Numerics, SCAN2016, Sep. 26-29, 2016.

[10] Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi: A norm estimation for an inverse of linear operator using a minimal eigenvalue, Computer Arithmetic and Validated Numerics, SCAN2016, Sep. 26-29, 2016.

[11] Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, and Shin’ichi Oishi: Verified numerical computations for blow-up solutions of ODEs, Computer Arithmetic and Validated Numerics, SCAN2016, Sep. 26-29, 2016.

㹌㹭

᪩

᪩✄⏣኱Ꮫ ༤ኈ㸦ᕤᏛ㸧 Ꮫ఩⏦ㄳ ◊✲ᴗ⦼᭩

✀㢮ู 㢟ྡࠊ Ⓨ⾲࣭Ⓨ⾜ᥖ㍕ㄅྡࠊ Ⓨ⾲࣭Ⓨ⾜ᖺ᭶ࠊ 㐃ྡ⪅㸦⏦ㄳ⪅ྵࡴ㸧

ㅮ₇

[12] Kazuaki Tanaka, Shin'ichi Oishi: On verified numerical computation for elliptic Dirichlet boundary value problems using sub- and super-solution method, The fifth Asian conference on Nonlinear Analysis and Optimization, Toki Messe, Niigata, Japan, August 1-6, 2016.

[13] Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi: Estimation for optimal constant satisfying an inequality for linear operator using minimal eigenvalue, The fifth Asian conference on Nonlinear Analysis and Optimization, Toki Messe, Niigata, Japan, August 1-6, 2016.

[14] Kaname Matsue, Akitoshi Takayasu, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, and Shin’ichi Oishi: Rigorous numerics of blowup solutions for ODEs, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, July 1-5, 2016.

[15] Kazuaki Tanaka, Kouta Sekine, Shin'ichi Oishi: Numerically verifiable condition for positivity of solution to elliptic equation, The 11th East Asia SIAM. June, 20-22, 2016.

[16] 㧗Ᏻு⣖, ᯇỤせ, బࠎᮌከᕼᏊ, ⏣୰୍ᡂ, Ỉཱྀಙ, ኱▼㐍୍㸸ᨺ≀㠃ࢥࣥࣃࢡࢺ

໬ࢆ⏝࠸ࡿᖖᚤศ᪉⛬ᘧࡢ⇿Ⓨゎࡢᩘ್ⓗ᳨ドἲ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2015ᖺᗘ㐃ྜ

Ⓨ⾲఍㸪⚄ᡞᏛ㝔኱Ꮫ࣏࣮ࢺ࢔࢖ࣛࣥࢻ࢟ࣕࣥࣃࢫ㸪2016ᖺ3᭶4᪥㹼3᭶5᪥㸬

[17] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, Shin'ichi Oishi: Numerical verification for positiveness of solutions to self-adjoint elliptic problems, JSST 2015 International Conference on Simulation Technology, Oct. 12-14, 2015.

[18] Kazuaki Tanaka and Shin'ichi Oishi: Computer-assisted analysis for solutions to nonlinear elliptic Neumann problems, JSST 2014 International Conference on Simulation Technology, Oct. 29-31, 2014.

[19] Kazuaki Tanaka and Shin'ichi Oishi: Numerical verification for periodic stationary solutions to the Allen-Cahn equation, The16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN2014, Sep. 21-26, 2014.

[20] 㧗Ᏻு⣖, ᯇỤせ㸪బࠎᮌከᕼᏊ㸪⏣୰୍ᡂ㸪Ỉཱྀಙ㸪኱▼㐍୍㸸Verified numerical

enclosure of blow-up time for ODEs㸪᪥ᮏᩘᏛ఍2015ᖺᗘᖺ఍㸪ி㒔⏘ᴗ኱Ꮫ㸪2015 ᖺ9᭶13᪥㹼9᭶16᪥㸬

[21] ⏣୰୍ᡂ, 㛵᰿᫭ኴ, Ỉཱྀಙ, ኱▼㐍୍㸸ᴃ෇ᆺ೫ᚤศ᪉⛬ᘧࡢゎࡢṇ್ᛶ࡟ᑐࡍࡿ

ᩘ್ⓗ᳨ドἲ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2015ᖺᗘᖺ఍㸪㔠ἑ኱Ꮫ㸪2015ᖺ9᭶9᪥㹼9᭶ 11᪥㸬

[22] ⱝᒣ㤾ኴ, ⏣୰୍ᡂ, 㛵᰿᫭ኴ, ᑿᓮඞஂ, ኱▼㐍୍㸸㏲ḟῧຍἲ࡟ࡼࡿ୕ゅᙧศ๭

ࡢDelaunay ᛶ࡟ᑐࡍࡿᩘ್ⓗ᳨ドἲ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2015ᖺᗘᖺ఍㸪㔠ἑ኱Ꮫ㸪

2015ᖺ9᭶9᪥㹼9᭶11᪥㸬

[23] 㧗Ᏻு⣖, ᯇỤせ㸪బࠎᮌከᕼᏊ㸪⏣୰୍ᡂ㸪Ỉཱྀಙ㸪኱▼㐍୍㸸ᖖᚤศ᪉⛬ᘧࡢ

⇿Ⓨゎ࡟ᑐࡍࡿ⢭ᗘಖド௜ࡁᩘ್ィ⟬㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2015ᖺᗘᖺ఍㸪㔠ἑ኱Ꮫ㸪 2015ᖺ9᭶9᪥㹼9᭶11᪥㸬

㹌㹭

᪩

᪩✄⏣኱Ꮫ ༤ኈ㸦ᕤᏛ㸧 Ꮫ఩⏦ㄳ ◊✲ᴗ⦼᭩

✀㢮ู 㢟ྡࠊ Ⓨ⾲࣭Ⓨ⾜ᥖ㍕ㄅྡࠊ Ⓨ⾲࣭Ⓨ⾜ᖺ᭶ࠊ 㐃ྡ⪅㸦⏦ㄳ⪅ྵࡴ㸧

ㅮ₇

ࡑࡢ௚

㸦 ࣏ ࢫ ࢱ

࣮Ⓨ⾲㸧

ࡑࡢ௚

㸦ཷ㈹㸧

[24] 㛵᰿᫭ኴ㸪⏣୰୍ᡂ㸪㧗Ᏻு⣖㸪ᒣᓮ᠇㸸ࢩࢢ࣐ࣀ࣒ࣝࢆ฼⏝ࡋࡓ⢭ᗘಖド௜ࡁᩘ

್ィ⟬ἲࡢ㐃❧ᴃ෇ᆺ೫ᚤศ᪉⛬ᘧ࡬ࡢᛂ⏝㸪➨㸲㸵ᅇ᪥ᮏ኱Ꮫ⏕⏘ᕤᏛ㒊Ꮫ⾡ㅮ

₇఍㸪᪥ᮏ኱Ꮫ㸪2014ᖺ12᭶6᪥㸬

[25] ⏣୰୍ᡂ㸪Ỉཱྀಙ㸪㛵᰿᫭ኴ㸪኱▼㐍୍㸸An a priori estimation of the Sobolev embedding constant and its application to numerical verification for solutions to PDEs, ➨10ᅇ᪥ᮏᛂ

⏝⌮Ꮫ఍◊✲㒊఍㐃ྜⓎ⾲఍㸪ி㒔኱Ꮫྜྷ⏣࢟ࣕࣥࣃࢫ⥲ྜ◊✲㸶ྕ㤋㸪 2014ᖺ3

᭶19᪥㹼3᭶20᪥㸬

[26] Kazuaki Tanaka and Shin'ichi Oishi: Numerical verification for stationary solutions to the Allen-Cahn equation, The International Workshop on Numerical Verification and its Applications, Waseda Univ. Nishiwaseda campus, Japan, March, 2014.

[27] Kazuaki Tanaka, Makoto Mizuguchi, Kouta Sekine, Akitoshi Takayasu, Shin'ichi Oishi:

Estimation of an embedding constant on Lipschitz domains using extension operators, JSST 2013 International Conference on Simulation Technology. Sep. 11-13, 2013.

[28] ⏣୰୍ᡂ, 㧗Ᏻு⣖, ๽㞷ᓠ, ኱▼㐍୍㸸⥺ᙧᴃ෇ᆺస⏝⣲ࡢNeumann᮲௳ୗ࡟࠾ࡅ

ࡿ⢭ᗘಖド௜ࡁ㏫స⏝⣲ࣀ࣒ࣝホ౯, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍ 2013ᖺᗘᖺ఍, ࢔ࢡࣟࢫ⚟

ᒸ, ⚟ᒸ┴⚟ᒸᕷ, 2013ᖺ9᭶9᪥㹼9᭶11᪥.

[29] Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu, Shin'ichi Oishi: Verified norm estimation for the inverse of linear elliptic operators and its application, The 9th East Asia SIAM. June 18-20, 2013.

[30] ⏣୰୍ᡂ, 㧗Ᏻு⣖, ๽㞷ᓠ, ኱▼㐍୍㸸㏫స⏝⣲ࣀ࣒ࣝホ౯ࢆ⏝࠸ࡓᴃ෇ᆺ

Neumannቃ⏺್ၥ㢟ࡢゎ࡟ᑐࡍࡿ⢭ᗘಖド௜ࡁᩘ್ィ⟬, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2013ᖺ

ᗘ㐃ྜⓎ⾲఍, ᮾὒ኱Ꮫⓑᒣ࢟ࣕࣥࣃࢫ, 2013ᖺ3᭶14᪥㹼3᭶15᪥.

[31] ⏣୰୍ᡂ, 㧗Ᏻு⣖, ๽㞷ᓠ, ኱▼㐍୍㸸⥺ᙧᴃ෇ᆺస⏝⣲ࡢNeumann᮲௳ୗ࡟࠾ࡅ

ࡿ⢭ᗘಖド௜ࡁ㏫స⏝⣲ࣀ࣒ࣝホ౯, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍ 2012ᖺᗘᖺ఍, ⛶ෆ඲᪥✵

࣍ࢸࣝ, ໭ᾏ㐨⛶ෆᕷ, 2012ᖺ8᭶28᪥㹼9᭶2᪥. 㸦ࡑࡢ௚ㅮ₇㸳௳㸧

[32] ⱝᒣ㤾ኴ, ⏣୰୍ᡂ, 㛵᰿᫭ኴ, ᑿᓮඞஂ, ኱▼㐍୍㸸Delaunay ୕ゅᙧศ๭ࡢ⢭ᗘಖ

ド௜ࡁᩘ್ィ⟬ᡭἲ࡟ᑐࡍࡿ⪃ᐹ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2016ᖺᗘᖺ఍㸪໭஑ᕞᅜ㝿఍

㆟ሙ㸪2016ᖺ9᭶12᪥㹼9᭶14᪥㸬 㸦ࡑࡢ௚࣏ࢫࢱ࣮Ⓨ⾲㸰௳㸧

[33] ⱝᒣ㤾ኴ, ⏣୰୍ᡂ, 㛵᰿᫭ኴ, ᑿᓮඞஂ, ኱▼㐍୍, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ఍2016ᖺᗘᖺ

఍㸪ඃ⚽࣏ࢫࢱ࣮㈹ ཷ㈹㸬

[34] ⏣୰୍ᡂ㸪2016ᖺᗘ኱ᕝຌグᛕ≉ูඃ⚽㈹ ཷ㈹㸬

[35] Kazuaki Tanaka, JSST 2014 International Conference, Student Presentation Award ཷ㈹㸬

[36] Kazuaki Tanaka, JSST 2013 International Conference, Student Presentation Award ཷ㈹㸬