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Partielle Differentialgleichungen: Sobolevräume Und Randwertaufgaben by Joseph Wloka

Partielle Differentialgleichungen: Sobolevräume Und Randwertaufgaben

by Joseph Wloka
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Book cover type: Paperback
  • ISBN13: 9783519022251
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Vieweg+teubner Verlag
  • Publisher Imprint: Vieweg+teubner Verlag
  • Publication Date:
  • Pages: 500
  • Original Price: EUR 51.39
  • Language: German
  • Edition: Softcover Repri
  • Item Weight: 799 grams
  • BISAC Subject(s): Engineering (General)

Gegenstand dieses Buches sind Randwertaufgaben. Für elliptische Differentialopera- toren werden durch die Bedingung von Lopatinskij-Sapiro ( = covering condition) alle Randwertbedingungen angegeben, die zur normalen Lösbarkeit eines Randwertpro- blems führen. Auch wird die Variationsmethode ausführlich dargelegt und Fragen nach ihrem Verhältnis zur allgemeinen elliptischen Theorie behandelt. Bei paraboli- schen und hyperbolischen Differentialoperatoren werden solche betrachtet, deren rechte Seite (Ableitungen nach x) ein elliptischer Differentialoperator ist, und die Kenntnisse über elliptische Operatoren werden benutzt, um Einsichten in die Lösbarkeit und die Regularitätseigenschaften der Lösung beim gemischten Problem zu gewinnen. Für die Bedingung von Lopatinskij-Sapiro habe ich eine Form gewählt, die es erlaubt, sofort zu testen, ob gegebene Randwertbedingungen sie erfüllen oder nicht. Es zeigt sich, da alle klassischen Randwertaufgaben ihr genügen, die Beispiele sind im Einzelnen nachgerechnet. Um den Umfang des Buches nicht zu sehr anschwellen zu lassen und den einführenden Charakter zu wahren, habe ich Pseudodifferentialoperatoren nicht behandelt; doch habe ich den Hauptsatz für elliptische Randwertprobleme durch Pseudodifferentialo- peratoren bewiesen - ohne sie so zu benennen. Den Differentialgleichungen habe ich ein ausführliches Kapitel über Distributionen und Sobolevräume vorangestellt; ich bin hier elementar vorgegangen, habe mit der Fouriertransformation gearbeitet und habe keine Interpolationssätze benutzt. Dies ist 2 - solange man innerhalb der L-Theorie bleibt - ohne weiteres möglich. Die U- Theorie habe ich nicht behandelt, sie bekommt ihr volles Gewicht erst für nichtlineare Gleichungen, siehe z. B. Lions [3], während sie für lineare nicht viele grundlegend neue Erkenntnisse bringt.

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