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Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces

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Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Dissertation zur Erlangung des Doktorgrades der Fakultat¨ fur¨ Mathematik der Universitat¨ Bielefeld vorgelegt von Jonas M. Tolle¨ im Januar 2010 Gedruckt auf alterungsbest¨andigemPapier nach DIN-ISO 9706. iii iv Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik der Universit¨atBielefeld vorgelegt von Dipl. Math. Jonas M. T¨olle aus Bielefeld im Januar 2010 1. Gutachter: Prof. Dr. Michael R¨ockner 2. Gutachter: Prof. Dr. Yuri G. Kondratiev Tag der m¨undlichen Pr¨ufung:5. Februar 2010 v vi Abstract In this doctoral thesis, a new approach towards variational convergence of quasi-linear monotone partial differential operators is elaborated. To this end, we analyze more ex- plicitly the so-called Kuwae-Shioya-convergence of metric spaces in the case of Banach spaces. For the first time, weak Banach space topologies are included. We achieve the objective to be able to formulate reasonable (topological) statements about convergence of vectors, functionals or operators such that each element of a convergent sequence lives in/on another distinct Banach space. Banach space-convergence is considered a natural generalization of Gromov-Hausdorff-convergence of compact metric spaces. The associ- ated theory is developed here completely and justified by several examples. Among other pn things, we are able to consider varying L (Ωn; Fn; µn)-spaces such that the measurable space (Ωn; Fn) as well as the measure µn as well as the degree of integrability pn varies for positive integers n, and such that the limit n ! 1 is given sense. Inside the framework of varying spaces, we show that a number of classical results on variational convergence still hold. Explicit applications are given for the equivalence of so-called Mosco-convergence of convex functionals and the strong-graph-convergence of the associated subdifferential operators. In the case of abstract Lp-spaces, we prove an elaborate result yielding a general transfer-method that enables us to carry over classical results (for one fixed space) to the case of varying spaces. More precisely, we construct isometries that respect the asymptotic topology of the varying Banach spaces and allow us to transform back to one fixed Banach space. We are considering four types of quasi-linear partial differential operators mapping a Banach space X to its dual space X∗. All of these operators are characterized completely via variational methods by lower semi-continuous convex functionals on a Banach space V embedded properly into X. As operators to be approximated, we present the weighted (non-homogeneous) Φ-Laplacian in Rd, the weighted p-Laplacian in Rd, the 1-Laplacian with vanishing trace in a bounded domain, and the generalized porous medium resp. fast diffusion operator in an abstract measure space. When taking the Mosco-approximation of their energies, we generally vary the weights (measures). In the second and third case, p is also varied. When dealing with approximations of such kind, varying spaces occur naturally. In the theory of homogenization, the special case of two-scale convergence has already been being employed for some time. Furthermore, we develop an alternative approach towards weighted p-Sobolev spaces of first order, which enables us to consider weights in a class different from the Muckenhoupt class. We prove a new result on density of smooth functions in weighted p-Sobolev spaces, which is known and well-studied as \Markov uniqueness" for p = 2. This problem is also known as \H = W ", that is, the coincidence of the strong and the weak Sobolev space. With the help of this result, we are able to identify the Mosco-limit of weighted p-Laplace operators. vii viii Zusammenfassung In dieser Dissertation wird ein neuer Zugang zur Variationskonvergenz von quasi-linearen monotonen partiellen Differenzialoperatoren erarbeitet. Zu diesem Zwecke wird die so genannte Kuwae-Shioya-Konvergenz von metrischen R¨aumen genauer im Banach-Raum- Fall untersucht. Erstmals werden schwache Banach-Raum-Topologien mit eingebunden. Wir erfullen¨ das Ziel, sinnvolle (topologische) Aussagen uber¨ Konvergenz von Vektoren, Funktionalen und Operatoren treffen zu k¨onnen, so dass jedes Element einer konvergen- ten Folge in oder auf einem ausgezeichneten, von den anderen verschiedenen Banach- Raum definiert ist. Banach-Raum-Konvergenz wird als naturliche¨ Verallgemeinerung der Gromov-Hausdorff-Konvergenz von kompakten metrischen R¨aumen angesehen. Die zugeh¨orige Theorie wird von uns vollst¨andig entwickelt und mit etlichen Beispielen ge- pn rechtfertigt. Wir sind unter anderem in der Lage, variierende L (Ωn; Fn; µn)-R¨aume zu betrachten, so dass sowohl der messbare Raum (Ωn; Fn), als auch das Maß µn sowie der Integrierbarkeitsgrad pn fur¨ naturliche¨ Zahlen n variieren, und so dass der Grenzubergang¨ n ! 1 sinnvoll ist. Im Rahmen der variierenden R¨aume zeigen wir, dass einige klassische Resultate uber¨ Variationskonvergenz weiterhin gelten. Genaue Anwendung fur¨ konkrete Operatoren fin- det die Aquivalenz¨ der so genannten Mosco-Konvergenz von konvexen Funktionalen und der Konvergenz der zugeh¨origen Subdifferenzialoperatoren im starken Graphen Sinne. Im Fall von abstrakten Lp-R¨aumen beweisen wir ein aufw¨andiges Resultat, in dem Isome- trien konstruiert werden, welche die asymptotische Topologie der variierenden Banach- R¨aume respektieren und es erlauben, eine Folge von Banach-R¨aumen auf einen festen Banach-Raum zuruckzutransformieren.¨ Wir betrachten vier verschiedene Typen von quasi-linearen partiellen Differentialope- ratoren, welche einen Banach-Raum X in dessen Dualraum X∗ abbilden. S¨amtliche dieser Operatoren werden vollst¨andig durch variationelle Methoden mittels unterhalbstetiger konvexer Funktionale auf einem in X echt eingebetteten Banach-Raum V beschrieben. Als zu approximierende Operatoren pr¨asentieren wir den gewichteten (nicht-homogenen) Φ-Laplace Operator in Rd, den gewichteten p-Laplace Operator in Rd, den 1-Laplace Operator mit verschwindender Spur in einer beschr¨ankten Dom¨ane und den verallgemei- nerten por¨ose Medien- bzw. schnelle Diffusions-Operator in einem abstrakten Maßraum. Bei der Mosco-Approximation derer Energien werden generell die Gewichte (Maße) und im zweiten und dritten Fall p variiert. Bei Approximationen dieser Art treten variierende R¨aume naturlicherweise¨ auf. In der Theorie der Homogenisation findet dies bereits seit einiger Zeit in dem Spezialfall der Zwei-Skalen-Konvergenz Anwendung. Weiterhin entwickeln wir einen alternativen Zugang zu gewichteten p-Sobolev-R¨aumen der ersten Ordnung, der uns eine Klasse von Gewichten einsetzen l¨asst, die sich von der Muckenhoupt-Klasse unterscheidet. Wir zeigen ein neues Resultat uber¨ Dichtheit glatter Funktionen in gewichteten p-Sobolev-R¨aumen, welches fur¨ p = 2 als Markoff- " Eindeutigkeit\ bekannt und wohlstudiert ist. Dieses Problem ist auch als H = W\ " bekannt; das Zusammenfallen des starken und schwachen Sobolev-Raumes. Mit Hilfe dieses Resultats sind wir in der Lage, den Mosco-Grenzwert von gewichteten p-Laplace Operatoren zu identifizieren. ix x Acknowledgements First of all, I would like to express my gratitude to Prof. Dr. Michael R¨ockner, who is the supervisor of this doctoral thesis and to whom I am deeply indebted. Without his personal support and kind patience, motivating discussions, clear insight into the problems involved and formulation of precise questions, this work would not have been possible. I can hardly stress enough how grateful I am for the fantastic opportunity to be a part of the Beijing{Bielefeld Graduate College, to study and do research both in Ger- many and China and to have had the possibility to attend many interesting workshops, conferences and summer schools all over the world. For advice, support and hospitality during my stays in Beijing and Wuhan, I would like to thank my \Chinese-side advisor" Prof. Dr. Ma Zhi-Ming. Furthermore, I would like to thank the second referee of this work, Prof. Dr. Yuri G. Kondratiev. For taking their time for my defense, I would like to thank Prof. Dr. Dr. h.c. Claus Michael Ringel and Prof. Dr. Wolf-J¨urgenBeyn. For fruitful discussions and proofreading the manuscript, a great thanks goes to Dr. Oleksandr Kutovyi. For offering a number of important remarks, I would like to thank the anonymous referee of my preprint from 2007. For proofreading my posters, I would like to thank Prof. Dr. Ludwig Streit. For helpful assistance and enduring kindness, I would like to thank the faculty adminis- trators Nicole Zimmermann, Gaby Windhorst, Nadja Epp, Hanne Litschewsky, Margret Meyer-Heidemann and Waltraud Bartsch. Also, a big thanks goes to Sven Wiesinger for his continuing and by all means incredible efforts. For guaranteeing a pleasant stay in Beijing, I would like to thank the staff member of the Chinese Academy of Sciences, Zhai Xiaoyun. For love, unshaken support and constant encouragement, I am deeply grateful to my family, in particular, to my parents Soile and Norbert, to my brother Felix and to my cousin Benjamin. For infinite helpfulness and hospitality in China, many inspiring discussions about Chinese culture and mathematics and their friendship, I am kindly asking Jin Peng, Liu Wei and Ouyang Shun-Xiang to accept my best thanks. I am gratefully paying tribute to my friends and companions Ang´elicaYohana Pach´on Pinz´on,Felipe Torres Tapia,Lukasz Derdziuk, N^azımHikmet Tekmen, Eko Nugroho, Sven Struckmeier, who visited with me. Thanks to Christiane for sending me a motivating postcard from each conference she has visited, and thanks to Christin for her backup and invaluable advice. Also, I am grateful to Michael for lending an ear or two from time to time in our kitchen. Last, not least, thanks to Verena for her excellent company in a laborious year 2009. Financial support by the DFG International Graduate College 1132 \Stochastics and Real World Models" Beijing{Bielefeld, the Graduate College of the Chinese Academy of Sciences, Beijing, and the Faculty of Mathematics, Universit¨atBielefeld, are gratefully acknowledged. Jonas M.
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