Quasilinear parabolic problems with nonlinear boundary conditions

More documents

Recommendations

Info

we have to show that there exist c > 0 and η ∈ (0, π/2) such that the inequality � � � � � � 1 � � � + 2� � 1 â(z)τ 2 � ≤ c � + �â(z)τ 2 � � 1 â(z)τ 2 + 1 4ˆb(z)+ 4 3 â(z) ˆ 4 b(z)+ 3 â(z) � 1 â(z)τ 2 + 1 + � 1 ( ˆb(z)+ 4 3 � � � � � + 1� â(z))τ 2 � , (z, τ) ∈ Σ π 2 +η × Ση holds true; here Σθ = {λ ∈ C \ {0} : |arg λ| < θ}. This crucial estimate is obtained by a careful function theoretic analysis. Finally, in Chapter 6 we study the nonlinear problem (1.1) described at the beginning and prove the last main result of the present thesis, Theorem 6.1.2, by means of the contraction mapping principle employing the optimal regularity results obtained for the linear problem (1.19). Section 6.1 deals with the fixed point construction and the basic estimates. It also contains the list of all assumptions needed for our treatment of (1.1). The proof of the harder estimates concerning in particular the nonlinearities on the boundary is deferred to Section 6.2. Acknowledgements. In the first place, I would like to express my gratitude to my supervisor, Prof. Dr. Jan Prüss. He is always open for discussing problems and an excellent teacher to me. I am also indebted to him for the participation in numerous national and international workshops and conferences. I am grateful to my colleagues, PD Dr. Roland Schnaubelt and Dipl.-Math. Matthias Kotschote, for many fruitful discussions and valuable suggestions. I would like to thank the Studienstiftung des deutschen Volkes, Bonn, for financial and non-material support. The thesis was also partially financially supported by a grant of the Graduiertenförderung des Landes Sachsen-Anhalt. I am grateful to Dr. Nina Grosser who critically and carefully read the manuscript of this work. I cannot forget all my friends who accompanied me during the last years. Finally, I would like to express my most sincere thanks to my parents for their constant support in every respect. 10
Chapter 2 Preliminaries 2.1 Some notation, function spaces, Laplace transform In this section we fix some of the notations used throughout this thesis, recall some basic definitions and give references concerning function spaces and the Laplace transform. By N, Z, R, C we denote the sets of natural numbers, integers, real and complex numbers, respectively. Let further N0 = N∪{0}, R+ = [0, ∞), C+ = {λ ∈ C : Re λ > 0}. X, Y, Z will usually be Banach spaces; | · |X designates the norm of the Banach space X. The symbol B(X, Y ) means the space of all bounded linear operators from X to Y , we write B(X) = B(X, X) for short. If A is a linear operator in X, D(A), R(A), N (A) stand for domain, range, and null space of A, respectively, while ρ(A), σ(A) designate resolvent set and spectrum of A. For a closed operator A we denote by DA the domain of A equipped with the graph norm. In what follows let X be a Banach space. For Ω ⊂ Rn open or closed, C(Ω; X) and BUC(Ω; X) stand for the continuous resp. bounded uniformly continuous functions f : Ω → X. Further, if Ω ⊂ Rn is open and k ∈ N, Ck (Ω; X) (BUC k (Ω; X)) designates the space of all functions f : Ω → X for which the partial derivative ∂αf exists on Ω and can be continuously extended to a function belonging to C(Ω; X) (BUC(Ω; X)), for each 0 ≤ |α| ≤ k. If Ω is a Lebesgue measurable subset of Rn and 1 ≤ p < ∞, then Lp(Ω; X) denotes the space of all (equivalence classes of) Bochner-measurable functions f : Ω → X with |f|p := ( � Ω |f(y)|p X dy)1/p < ∞. Lp(Ω; X) is a Banach space when normed by | · |p. For an interval J ⊂ R, s > 0 and 1 < p < ∞, by Hs p(J; X) and Bs pp(J; X) we mean the vector-valued Bessel potential space resp. Sobolev-Slobodeckij space of Xvalued functions on J, see Amann [6], Schmeisser [73], ˇ Strkalj [76], and Zimmermann [83]. Concerning the scalar case, we refer further to Runst and Sickel [72], Triebel [78], [79]. In Section 2.8 we give a definition of the spaces Hs p(J; X) in the situation where X belongs to the class HT (cf. Section 2.3); this will always be the case when we are to consider vector-valued Bessel potential spaces. We will frequently use the property that the Sobolev-Slobodeckij spaces appear as real interpolation spaces between the spaces Lp and Hs p; more precisely (Lp(J; X), Hs p(J; X))θ, p = Bθs pp(J; X) for all 1 0, and θ ∈ (0, 1). For general treatises on interpolation theory we refer to Bergh and Löfström [9], and Triebel [78]. If F is any of the above function spaces, then f ∈ Floc means that f belongs to the corresponding space when restricted to compact subsets of its domain. In the scalar case X = R or X = C we usually omit the image space in the function space notation. 11
Page 1 and 2: Gutachter: Quasilinear parabolic pr
Page 3 and 4: Contents 1 Introduction 3 2 Prelimi
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: conditions of order ≤ 1. Sections
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
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 and 106:
substitution operators to be studie
Page 107 and 108:
for all t, τ ∈ J, ξ, η ∈ K,
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
show all

Quasilinear parabolic problems with nonlinear boundary conditions

Create successful ePaper yourself

Delete template?

Save as template?