Some principles of analysis

Jürgen Jost
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Jürgen Jost: Ruhr-Universität Bochum, Mathematisches Institut

Chapter 2 in Nonlinear Methods in Riemannian and Kählerian Geometry, 1991, pp 73-86 from Springer

Abstract: Abstract Let L t : B 1 → B 2, B 1, B 2, Banachspaces be a family of (partial differential) operators, depending smoothly on a parameter t; typically t ∈ [0, 1] or t ∈ [0, ∞), and one knows a solution u 0 for t = 0, i.e. % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa % aaleaacaaIWaaabeaakiaadwhadaWgaaWcbaGaaGimaaqabaGccqGH % 9aqpcaaIWaGaaiilaaaa!3C0F! $${L_0}{u_0} = 0,$$ and one wants to find a solution u t % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa % aaleaacaWG0baabeaakiaadYeadaWgaaWcbaGaamiDaaqabaGccqGH % 9aqpcaaIWaaaaa!3BB4! $${L_t}{L_t} = 0$$ for all t, in particular either for t = 1 or for t → ∞, and in the latter case one would like to have convergence of u t as t → ∞. The proof usually consists of two steps; namely one shows that % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeIuUaai % Ooaiabg2da9maacmaabaGaamiDaiaacQdacqGHdicjcaWG1bWaaSba % aSqaaiaadshaaeqaaOGaaGjcVlaabEhacaqGPbGaaeiDaiaabIgaca % aMi8UaamitamaaBaaaleaacaWG0baabeaakiaadwhadaWgaaWcbaGa % amiDaaqabaGccqGH9aqpcaaIWaaacaGL7bGaayzFaaaaaa!4D15! $$\sum : = \left\{ {t:\exists {u_t}{\kern 1pt} {\text{with}}{\kern 1pt} {L_t}{u_t} = 0} \right\}$$ is both open and closed (in [0, 1] or [0, ∞), resp.). Since by assumption 0 ∈ Σ, one concludes that a solution exists for every t.

Keywords: Riemannian Manifold; Parabolic Equation; Sectional Curvature; Implicit Function Theorem; Compact Riemannian Manifold (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-3-0348-7706-0_2

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