This thesis is dedicated to the so-called Dirichlet-to-Neumann map associated with the weighted ๐-Laplace operator. In Chapter 1, we begin by deriving the Dirichlet-to-Neumann map by using classical modelling and outline why it is interesting to study this boundary operator. In the remaining part of Chapter 1, we dedicate each section an overview about the content of one chapter and summarize the main results. Chapter 2 is dedicated to the Poisson problem and the inverse of the Dirichlet-to-Neumann map. Chapter 3 provides the first main application of the Dirichlet-to-Neumann map, namely, it generates a strongly continuous semigroup of contractions on the Lebesgue space ๐ฟ2 and this contraction can be extrapolated to a contraction on ๐ฟq for all 1 โค ๐ โค โ. In Chapter 4, we develop an abstract theory to establish global ๐ฟ๐-๐ฟโ regularization estimates satisfied by the semigroup generated by the negative Dirichlet-to-Neumannmap. Chapter 5 is concerned with ๐ฟ1 and point-wise estimates on the time-derivative of the semigroup generated by the neagtive Dirichlet-to- Neumann map, which are known in the literatur as Aronson-Beฬnilan type estimates. In Chapter 6, we outline the theory of ๐-functional and its application to evolution problems. This theory allows us to study the Dirichlet problem on general open sets ฮฉ, and to realize the Dirichlet-to-Neumann map as an operator in ๐ฟ2 (๐ฮฉ). In Chapter 7, we consider the limit case ๐ = 1, which corresponds to the Dirichlet-to-Neumann map associated with the (unweighted) 1-Laplace operator. Each chapter covers parts of the authors papers mentioned in the references.:Chapter 1
Introduction................................................... 1
1.1 Motivation-physical background ............................. 2
1.2 The Dirichlet-to-Neumann map - an analystโs perspective . . . . . . . . . 5
1.2.1 Step1. The Dirichlet problem.......................... 5
1.2.2 Step2. The Neumann boundary operator ................ 8 2
1.3 The Dirichlet-to-Neumann map on ๐ฟ2 ......................... 9
1.4 The Dirichlet-to-Neumann map and Leray-Lions operators . . . . . . . . 11
1.5 The Dirichlet-to-Neumann map is a nonlocal operator . . . . . . . . . . . . 12
1.6 The Dirichlet-to-Neumann map on open sets.................... 13
1.6.1 ๐-elliptic functionals and their ๐-subgradient . . . . . . . . . . . . . 13
1.6.2 The construction of a weak trace on open sets ............ 15
1.6.3 Construction of the Dirichlet-to-Neumann map . . . . . . . . . . . 17
1.7 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . 19
1.8 Evolution problems governed by the Dirichlet-to-Neumann map . . . 21
1.9 ๐ฟ๐-๐ฟโ regularization and decay estimates...................... 27
1.10 Aronson-Beฬnilantypeestimates .............................. 30
1.11 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . 33
Chapter 2
The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . . . 45
2.1 The Poisson problem........................................ 45
2.2 Preliminaries .............................................. 46
2.3 The Dirichlet problem....................................... 48
2.4 The Dirichlet-to-Neumann map............................... 51
2.5 Proof of Theorem 2.1 ....................................... 56
2.5.1 Proof of claim (1) of Theorem 2.1 ...................... 56
2.5.2 Preliminaries for the proof of claim (2) of Theorem 2.1 . . . . 58
2.5.3 Proof of claim( 2) of Theorem 2.1 ...................... 60
Chapter 3
Nonlinear elliptic-parabolic evolution problems.................... 61
3.1 Main result................................................ 61
3.2 Preliminaries .............................................. 64
3.2.1 Some function spaces................................. 64
3.2.2 Nonlinear semigroupt heory - Part I..................... 65
3.2.3 Homogeneous operators - Part I ........................ 75 2
3.3 The Dirichlet-to-Neumann map on ๐ฟ2 ...................... 77
3.4 The Dirichlet-to-Neumann map on ๐ฟ1, ๐ฟ๐ and C................ 82
3.5 Proof of Theorem 3.1 ....................................... 84
Chapter 4
๐ณ๐-๐ณโ regularization and decay estimates ........................ 89
4.1 Main results............................................... 89
4.2 Preliminaries .............................................. 91
4.3 Sobolev implies ๐ฟ๐ -๐ฟ๐ regularization estimates ................. 92
4.4 Extrapolation towards ๐ฟ1 .................................... 98
4.5 A nonlinear interpolation theorem.............................100
4.6 Extrapolation towards ๐ฟโ via interpolation of the semigroup . . . . . . 107
4.7 Proof of Theorem 4.1 .......................................115
Chapter 5
Aronson-Beฬnilan type estimates..................................117
5.1 Main results ...............................................117
5.2 Preliminaries ..............................................119
5.2.1 Nonlinearsemigrouptheory-PartII ....................119
5.2.2 Homogeneousaccretiveoperators ......................130
5.2.3 Homogeneous completely accretive operators . . . . . . . . . . . . 138
5.3 Proof of Theorem 5.1 .......................................141
Chapter 6
The Dirichlet-to-Neumann map on open sets ......................143
6.1 Main results ...............................................143
6.2 The ๐-subgradient and basic properties ........................146
6.2.1 Definition and characterisation as a classical gradient . . . . . . 146
6.2.2 Ellipticextensions ...................................151 H
6.2.3 Identification of ๐ ..................................152
6.2.4 The case when ๐ is a weakly closed operator .............155
6.3 Semigroups and invariance of convex sets ......................156
6.3.1 Positive semigroups ..................................160
6.3.2 Comparison and domination of semigroups ..............161
6.3.3 ๐ฟโ-contractivity and extrapolation of semigroups . . . . . . . . . 163
6.4 Application:The Dirichlet-to-Neumann map....................168
Chapter 7
The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . . . 171
7.1 Preliminaries ..............................................171
7.1.1 Functions of bounded variation.........................171
7.1.2 Nonlinear semigroup theory - Part III ...................178
7.2 The Dirichlet problem for the 1-Laplace operator................180
7.3 A Robin-type problem for the 1-Laplace operator................187
7.4 Proofs of the main results....................................189
7.4.1 The Dirichlet-to-Neumann operator in ๐ฟ1 ................189
7.4.2 The Dirichlet-to-Neumann operator in ๐ฟ2 ................200
7.4.3 The Dirichlet-to-Neumann operator in ๐ฟ1 (continued)...........204
7.4.4 Long-timestability...................................206
Appendix
A Weighted Sobolev Spaces........................................213
A.1 p-admissible weights........................................213
B Mean spaces by Lions and Peetre ................................215
B.1 The connection between mean spaces and ๐ฟp spaces.............215
References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 219
Index .............................................................227
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:91016 |
| Date | 22 April 2024 |
| Creators | Hauer, Daniel |
| Contributors | Chill, Ralph, Toledo, Julian Josรฉ, Barbu, Viorel, Technische Universitรคt Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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