Quasilinear parabolic problems with nonlinear boundary conditions

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Contents 1 Introduction 3 2 Preliminaries 11 2.1 Some notation, function spaces, Laplace transform . . . . . . . . . . . . . 11 2.2 Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Sums of closed linear operators . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Joint functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Operator-valued Fourier multipliers . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Evolutionary integral equations . . . . . . . . . . . . . . . . . . . . . . . . 27 2.8 Volterra operators in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Maximal Regularity for Abstract Equations 33 3.1 Abstract parabolic Volterra equations . . . . . . . . . . . . . . . . . . . . 33 3.2 A general trace theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 More time regularity for Volterra equations . . . . . . . . . . . . . . . . . 46 3.4 Abstract equations of first and second order on the halfline . . . . . . . . 47 3.5 Parabolic Volterra equations on an infinite strip . . . . . . . . . . . . . . . 48 4 Linear Problems of Second Order 57 4.1 Full space problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Half space problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.1 Constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.2 Pointwise multiplication . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.3 Variable coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Problems in domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Linear Viscoelasticity 79 5.1 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Assumptions on the kernels and formulation of the goal . . . . . . . . . . 81 5.3 A homogeneous and isotropic material in a half space . . . . . . . . . . . 82 5.3.1 The case δa ≤ δb: necessary conditions . . . . . . . . . . . . . . . . 83 5.3.2 The case δa ≤ δb: sufficiency of (N1) . . . . . . . . . . . . . . . . . 84 5.3.3 The case 0 < δa − δb < 1/p . . . . . . . . . . . . . . . . . . . . . . 91 5.3.4 The case δa − δb > 1/p . . . . . . . . . . . . . . . . . . . . . . . . . 92 1
Page 1: Gutachter: Quasilinear parabolic pr
Page 5 and 6: Chapter 1 Introduction The present
Page 7 and 8: to assume that m, c are bounded fun
Page 9 and 10: We give now an overview of the cont
Page 11 and 12: conditions of order ≤ 1. Sections
Page 13 and 14: Chapter 2 Preliminaries 2.1 Some no
Page 15 and 16: Clearly, φA ∈ [0, π) and φA
Page 17 and 18: Definition 2.2.3 Let X and Y be Ban
Page 19 and 20: µ ∈ Σφα }. Let N ∈ N, Tj
Page 21 and 22: We remark that a theorem of the Dor
Page 23 and 24: Here f(A, ·) ∈ H0(Σ π 2 +η; B
Page 25 and 26: Further, K ∞ (α, θa) := {a ∈
Page 27 and 28: Using (2.19) for aω and bω yields
Page 29 and 30: 2.7 Evolutionary integral equations
Page 31 and 32: Example 2.8.1 For J = [0, T ] and a
Page 33: We conclude this section by illustr
Page 36 and 37: kernel a. The operator B is inverti
Page 38 and 39: x := f(0) ∈ X exists and we are l
Page 40 and 41: with two positive constants C1, C2
Page 42 and 43: with two positive constants C1 and
Page 44 and 45: derivative theorem to this pair of
Page 46 and 47: 3.2 A general trace theorem Let X b
Page 48 and 49: 3.3 More time regularity for Volter
Page 50 and 51: Theorem 3.4.2 Suppose X is a Banach
Page 52 and 53:
Our next objective is to show neces
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Let u1 be the restriction of v1 to
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Proof. We begin with the necessity
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Chapter 4 Linear Problems of Second
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The strategy for solving (4.1) is n
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Since ψj ≡ 1 on supp ϕj, we may
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Turning to (c), let g ∈ Ξi+1 and
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endowed with the norm | · | Y T 2
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We remark that the constant C2 stem
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One can then construct functions a
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analogous to (4.17), shows that S i
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Apply now V#, i+1 := I + k ∗ A#(
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Given a function v ∈ H 2 p(R n+1
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v is a solution of (4.40) on Ji+1 :
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Chapter 5 Linear Viscoelasticity In
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where δij denotes Kronecker’s sy
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problem ⎧ ⎪⎨ ⎪⎩ ∂tv −
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To see the converse direction, supp
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and up solves � Aup − ∆xup
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It can be written as where l(z, ξ)
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elongs to H∞ (Σ π 2 +η × Ση
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which allows us to write the first
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Chapter 6 Nonlinear Problems 6.1 Qu
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sufficiently small, say T ≤ T1
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(d) bD ∈ C(J0 × ΓD × U0), ∃C
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which entails (6.14). Corresponding
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substitution operators to be studie
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for all t, τ ∈ J, ξ, η ∈ K,
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to write where h2(t, τ, x) = h21(t
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Lemma 6.2.3 Let 0 < s < s0 < 1, ρ
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Bibliography [1] Albrecht, D.: Func
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[54] Lunardi, A.: On the heat equat
Page 117 and 118:
kleines T mit Hilfe des Kontraktion
Page 119:
Personal Details Curriculum Vitae N
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Quasilinear parabolic problems with nonlinear boundary conditions

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