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Local analysis of Loewner equation

Abstract : This thesis studies the curve generation problem of the general Loewner equation. We use a local transformation in the chordal Loewner equation, and analyse the solution of the Loewner equation, obtain three results.At first, we analyse the Limit superior and limit inferior of the left 1/2 order of the driving function, then we prove a basic lemma about that the generation curves do not intersect with itself locally. By this lemma, we have three conclusion. Firstly, Lind proved that when 1/2-Hölder norm is less than 4, then the Loewner equation is generated by a simple curve. We discuss the case that the 1/2-Hölder norm is greater than 4, and give a sufficient condition of the generation curve is simple. Secondly, the limit inferior of the 1/2 order of the Brownian motion will tends to 0 locally, we give a estimation of the speed of it tends to 0. Thirdly, we proof that for the1/2 order Weierstrass function with coefficient less that a constant, the Loewner equation which is driven by it is generated by a simple curve.In the second part, we define the imaginary Loewner equation and its dual equation, and we do the local transformation for these two equation, after analyse their vanishing property, we build the connection between it with the curve generation problem. And then we give a sufficient condition on that the Loewner equation is generated by a curve locally.At last, we define and discuss the left self-similar driving function, and use the knowledge of complex dynamic to prove that if it is generated by a curve in the upper-half plane locally, then it is generated by a curve entirely.
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Henshui Zhang. Local analysis of Loewner equation. General Mathematics [math.GM]. Université d'Orléans, 2018. English. ⟨NNT : 2018ORLE2064⟩. ⟨tel-02988117⟩

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