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Inverse problems by Carleman estimates for several partial differential equations in mathematical physics

黄, 欣馳 東京大学 DOI:10.15083/0002003734

2022.04.20

概要

さまざまな自然現象の解明のためには数理モデルが必要である.多くの場合に,数理モデルは偏微分方程式で記述することができる.その際に,偏微分方程式の係数や非斉次項がわかっているものとすれば,適切に初期条件や境界条件を付ければ,それらを満たす解の一意性や存在さらには解の性質などを解析することができる.このような問題は順問題とよばれる.しかしながら,モデル化にあたっては,偏微分方程式の型はわかるが,係数などがあらかじめわかっていることは稀で解についての何らかのデータから推定しなくてはならないことが多い.これは,順問題と異なり,計測データ(結果)から係数(原因)を探る逆問題である.逆問題は定量的なモデル化のために必要であり,多様な分野で注目を集めている.本論文では,以下のような3つの発展方程式に関する逆問題の一意性及び安定性について考察した:

1.非圧縮性粘性流体に対する磁気流体力学方程式,
2.異常拡散現象に対する時間非整数階移流拡散方程式及び,
3.光学フォノン散乱のあるSchr¨odinger方程式.

本学位論文では,主たる結果はカーレマン評価という手法で証明された.カーレマン評価は偏微分方程式の解に対するパラメータを含む重み付きのL2評価で,最初にCarleman(1939)によって二次元の楕円型方程式に対する一意接続性の証明のために導入された.1980年代Bukhgeim-Klibanovにより,逆問題の一意性の証明のために応用された.それ以来,多くの研究者が,彼らの方法論を発展させて,放物型方程式や双曲型方程式などの単独の偏微分方程式の空間変数に依存する係数や非斉次項の決定逆問題に対して一意性にとどまらずに安定性の成果を確立してきた.

本論文では,逆問題の安定性を中心として論じた.主要結果を述べる前に,共通の記号を幾つか導入する.T>0であって,Ω⊂RNを十分滑らかな境界∂Ωをもつ有界領域とし,Q:=Ω×(0,T)とおく.しかも,nを単位法線ベクトルとして∂nを法線に沿った方向微分とする.さらに,任意の空でない部分境界Γに対してある関数d∈C2(Ω)が存在し,条件d>0inΩ,|∇d|>0onΩ,d=0on∂Ω\Γを満たす.その関数dにより,勝手に与えられたε>0に対し,次のようなΩの部分集合が定義できる:Ωε:={x∈Ω;d(x)>ε}.以下,本学位論文の主要結果について章ごとに述べる.

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