Quasilinear parabolic problems with nonlinear boundary conditions

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and which serves as a reference function in the following sense. For F = Y k1,s1 1 ∩ Y k2,s2 2 , and given ρ ∈ (0, ρ0], let Σ = Σ(ρ, w, F ) be the set of all K-valued f in F m such that f − w ∈ 0F m and |f − w|F m ≤ ρ. The aim is to show that Bf ∈ F whenever f ∈ Σ and that |Bf − Bg|F ≤ C(ρ + µ(T ) + |bξ(·, ·, w)|∞, m)|f − g|F m, f, g ∈ Σ, (6.19) where C > 0 is independent of ρ and T , and 0 < µ(T ) → 0 as T tends to zero. We further need the property that Bf ∈ X1, s if f ∈ X m 1, s0 ∩ (C0 1 )m (s0 ∈ (s, 1)), and we wish to have a Lipschitz estimate of the form |Bf − Bg|X1, s ≤ µ(T )(|f − g|X m 1, s 0 + |f − g|∞, m), (6.20) for all (K-valued) f, g ∈ Σ ′ = Σ ′ (ρ, w) := {h ∈ X m 1, s0 ∩ (C0 1 )m : (h − w)|t=0 = 0 and |h − w|X m 1, s0 + |h − w|∞, m ≤ ρ}, where again the constant µ(T ) > 0 vanishes as T → 0. In both cases we shall only establish the Lipschitz estimate. The corresponding mapping property of B follows by means of the same techniques; here the proof is even simpler than for the Lipschitz estimate. When proving Lipschitz estimates we will restrict ourselves to the seminorms of highest order, that is, e.g., if we are to estimate Bf − Bg in the Y 1,s1 1 and [Bf − Bg] Y 1,s 2 2 ∩ Y 1,s2 2 -norm, we shall consider only the seminorms [Bf − Bg] Y 1,s 1 1 . Having proved the desired estimate for these terms, it will then be clear how to obtain it for the seminorm terms of lower order, which are much easier to treat. We begin now with the spaces (Y k1,s1 1 ∩ Y k2,s2 2 ) m = B k1+s1 pp (J; Lp(Ω, R m )) ∩ Lp(J; B k2+s2 pp (Ω, R m )), ki ∈ {0, 1}, si ∈ (0, 1), i = 1, 2. Here, the cases (k1, k2) = (0, 0), (0, 1), (1, 1) have to be studied. We assume that in each of these cases, p is large enough such that we have the embedding Y 0,s1 1 ∩ Y 0,s2 2 ↩→ C(J; C( ¯ Ω)), Y 0,s1 1 ∩ Y 1,s2 2 ↩→ C(J; C1 ( ¯ Ω)), and Y 1,s1 1 ∩ Y 1,s2 2 ↩→ Cr (J; C( ¯ Ω)) ∩ C(J; C1 ( ¯ Ω)), respectively, with some number r in (s1, 1). More precisely, we make the assumption that p > p⋆(n, k1, s1, k2, s2), where ⎧ ⎪⎨ p⋆(n, k1, s1, k2, s2) := ⎪⎩ 1 s1 1 s1 max + n s2 (1 + 1 � 1 s2 s2 ) + n s2 ( 1+s2 1+s1 + n), 1 + n 1+s1 1+s2 � : (k1, k2) = (0, 0) : (k1, k2) = (0, 1) : (k1, k2) = (1, 1). Roughly speaking, this condition on the exponent p when translated to the situation of Section 6.1 corresponds to the assumption (H2) therein. The first lemma is concerned with the case (k1, k2) = (0, 0). Here and in the subsequent estimates we denote by µ(T ) a positive constant depending on T such that µ(T ) → 0 as T → 0. Further, M and C denote constants which may differ from line to line, but which do not depend on T and ρ. Lemma 6.2.1 Let w ∈ (Y 0,s1 1 ∩ Y 0,s2 2 ) m be a fixed K-valued function and ρ > 0. Let further Σ be as described above. Suppose that there exist CHL > 0, r ∈ (s1, 1), and Cb ∈ Lp(Ω) such that |bξ(t, x, ξ) − bξ(τ, x, η)| ≤ CHL(Cb(x)|t − τ| r + |ξ − η|), 104
for all t, τ ∈ J, ξ, η ∈ K, and a.a. x ∈ Ω. Then there exists a constant C > 0 not depending on T and ρ such that � � ≤ C ρ + µ(T ) + |bξ(·, ·, w)|∞, m [b(·, ·, f) − b(·, ·, g)] Y 0,s 1 1 Proof. Let f and g be arbitrary functions in Σ. Put |f − g| (Y 0,s 1 1 ∩Y 0,s2 2 ) m, f, g ∈ Σ. h(t, τ, x) = b(t, x, f(t, x)) − b(t, x, g(t, x)) − b(τ, x, f(τ, x)) + b(τ, x, g(τ, x)) for t, τ ∈ J, and a.a. x ∈ Ω. Then [b(·, ·, f) − b(·, ·, g)] Y 0,s 1 1 � T � T � = ( 0 0 Ω |h(t, τ, x)| p 1 dx dτ dt) p . |t − τ| 1+s1p Letting φ(t, τ, x, θ) = bξ(t, x, g(τ, x) + θ(f(τ, x) − g(τ, x))), t, τ ∈ J, x ∈ Ω, θ ∈ [0, 1], we write h(t, τ, x) = = � 1 0 � 1 0 � 1 + φ(t, t, x, θ) dθ · (f(t, x) − g(t, x))− � 1 φ(t, t, x, θ) dθ · (f(t, x) − f(τ, x) − g(t, x) + g(τ, x))+ 0 (φ(t, t, x, θ) − φ(τ, τ, x, θ)) dθ · (f(τ, x) − g(τ, x)). 0 φ(τ, τ, x, θ) dθ · (f(τ, x) − g(τ, x)) With ψ(f, g) := ess sup{|φ(t, t, x, θ)| : t ∈ J, x ∈ Ω, θ ∈ [0, 1]}, we therefore have |h(t, τ, x)| ≤ ψ(f, g)|f(t, x) − f(τ, x) − g(t, x) + g(τ, x)| + � � 1 + ((1 − θ)|g(t, x) − g(τ, x)| + θ|f(t, x) − f(τ, x)|) dθ + CHL 0 +CHLCb(x)|t − τ| r� · |f(τ, x) − g(τ, x)| ≤ ψ(f, g)|f(t, x) − f(τ, x) − g(t, x) + g(τ, x)| + CHL|f − g|∞ · ·(|f(t, x) − f(τ, x)| + |g(t, x) − g(τ, x)| + Cb(x)|t − τ| r ) for all t, τ ∈ J, and a.a. x ∈ Ω. On the whole we thus find that [b(·, ·, f) − b(·, ·, g)] Y 0,s 1 1 ≤ ψ(f, g)[f − g] (Y 0,s 1 1 +[g] 0,s (Y 1 1 ) m + |Cb| Lp(Ω)( = ψ(f, g)[f − g] (Y 0,s 1 1 +[g] 0,s (Y 1 1 ) m + |Cb| Lp(Ω)C1T where C1 = 2 1/p [(r − s1)p(1 + (r − s1)p)] −1/p . Y 0,s1 1 � [f] 0,s (Y 1 1 ) m + ) m + CHL|f − g|∞, m � T � T dτ dt � 1 ) p 0 0 |t − τ| 1+(s1−r)p � ) m + CHL|f − g|∞, m r−s1+ 1 p � , [f] 0,s (Y 1 1 ) m + By definition of Σ, the Lipschitz estimate for bξ, and in view of the embedding ↩→ C(J; C( ¯ Ω)), we have the inequalities ∩ Y 0,s2 2 [f − g]∞, m ≤ M|f − g| 0,s (Y 1 1 ∩Y 0,s2 2 ) m, [f] 0,s (Y 1 1 ) m + [g] 0,s (Y 1 1 ) m ≤ 2(ρ + [w] 0,s (Y 1 1 ) m), 105
Page 1 and 2:
Gutachter: Quasilinear parabolic pr
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Contents 1 Introduction 3 2 Prelimi
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Chapter 1 Introduction The present
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to assume that m, c are bounded fun
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We give now an overview of the cont
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conditions of order ≤ 1. Sections
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Chapter 2 Preliminaries 2.1 Some no
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Clearly, φA ∈ [0, π) and φA
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Definition 2.2.3 Let X and Y be Ban
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µ ∈ Σφα }. Let N ∈ N, Tj
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We remark that a theorem of the Dor
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Here f(A, ·) ∈ H0(Σ π 2 +η; B
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Further, K ∞ (α, θa) := {a ∈
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Using (2.19) for aω and bω yields
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2.7 Evolutionary integral equations
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Example 2.8.1 For J = [0, T ] and a
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We conclude this section by illustr
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kernel a. The operator B is inverti
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x := f(0) ∈ X exists and we are l
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with two positive constants C1, C2
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with two positive constants C1 and
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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
Page 54 and 55:
Let u1 be the restriction of v1 to
Page 56 and 57: Proof. We begin with the necessity
Page 59 and 60: Chapter 4 Linear Problems of Second
Page 61 and 62: The strategy for solving (4.1) is n
Page 63 and 64: Since ψj ≡ 1 on supp ϕj, we may
Page 65 and 66: Turning to (c), let g ∈ Ξi+1 and
Page 67 and 68: endowed with the norm | · | Y T 2
Page 69 and 70: We remark that the constant C2 stem
Page 71 and 72: One can then construct functions a
Page 73 and 74: analogous to (4.17), shows that S i
Page 75 and 76: Apply now V#, i+1 := I + k ∗ A#(
Page 77 and 78: Given a function v ∈ H 2 p(R n+1
Page 79 and 80: v is a solution of (4.40) on Ji+1 :
Page 81 and 82: Chapter 5 Linear Viscoelasticity In
Page 83 and 84: where δij denotes Kronecker’s sy
Page 85 and 86: problem ⎧ ⎪⎨ ⎪⎩ ∂tv −
Page 87 and 88: To see the converse direction, supp
Page 89 and 90: and up solves � Aup − ∆xup
Page 91 and 92: It can be written as where l(z, ξ)
Page 93 and 94: elongs to H∞ (Σ π 2 +η × Ση
Page 95 and 96: which allows us to write the first
Page 97 and 98: Chapter 6 Nonlinear Problems 6.1 Qu
Page 99 and 100: sufficiently small, say T ≤ T1
Page 101 and 102: (d) bD ∈ C(J0 × ΓD × U0), ∃C
Page 103 and 104: which entails (6.14). Corresponding
Page 105: substitution operators to be studie
Page 109 and 110: to write where h2(t, τ, x) = h21(t
Page 111 and 112: Lemma 6.2.3 Let 0 < s < s0 < 1, ρ
Page 113 and 114: Bibliography [1] Albrecht, D.: Func
Page 115 and 116: [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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