TY - THES A1 - Winkler, Ralf T1 - Schwache Randwertprobleme von Systemen elliptischen Charakters auf konischen Gebieten T1 - Weak boundary value problems of linear elliptic systems on conical domains N2 - In der vorliegenden Arbeit werden lineare Systeme elliptischer partieller Differentialgleichungen in schwacher Formulierung auf konischen Gebieten untersucht. Auf einem zunächst unbeschränkten Kegelgebiet betrachten wir den Fall beschränkter und nur von den Winkelvariablen abhängiger Koeffizientenfunktionen. Die durch selbige definierte Bilinearform genüge einer Gårdingschen Ungleichung. In gewichteten Sobolevräumen werden Existenz- und Eindeutigkeitsfragen geklärt, wobei das Problem mittels Fouriertransformation auf eine von einem komplexen Parameter abhängige Familie T(·) von Fredholmoperatoren zurückgeführt wird. Unter Anwendung des Residuenkalküls gewinnen wir eine Darstellung der Lösung in Form einer Zerlegung in einen glatten Anteil einerseits sowie eine endliche Summe von Singulärfunktionen andererseits. Durch Abschneidetechniken werden die gewonnenen Erkenntnisse auf den Fall schwach formulierter elliptischer Systeme auf beschränkten Kegelgebieten unter Formulierung in gewöhnlichen, nicht-gewichteten Sobolevräumen angewendet. Die für Regularitätsfragen maßgeblichen Eigenwerte der Operatorfunktion T mit minimalem positiven Imaginärteil werden im letzten Kapitel der Arbeit am Beispiel der ebenen elastischen Gleichungen numerisch bestimmt. N2 - In the present PhD thesis we investigate systems of linear partial elliptic equations in weak formulation on conical domains. For an unbounded cone, first, we study the case of bounded and radially constant coefficient functions. The so defined bilinear form is supposed to satisfy a (local) Gårding inequality. In weighted Sobolev spaces we study questions of existence and uniqueness of solutions. In this context the problem is Fourier-transformed onto a set of smaller problems, represented by Fredholm operators T(·) that holomorphically depend on a complex parameter. Via the residual theorem we yield a decomposition of the solution into a regular part and a finite sum of singular functions. Using cut-off techniques we are able to transfer the preceeding results onto the case of weak formulated linear elliptic systems on bounded cones under restriction to usual, non weighted Sobolev spaces. In the last chapter, the eigenvalues of T with minimal positive imaginary part, which are responsible for regularity properties, are numeriaclly determined for the example of the plane Elastic Equations. KW - Elliptische Differentialgleichung KW - Lineare Funktionalanalysis KW - Funktionentheorie KW - Numerische Mathematik KW - Kegelgebiet KW - unstetige Koeffizientenfunktionen KW - gewichtete Sobolevräume KW - Singulärfunktionen KW - conical domain KW - discontinuous coefficient functions KW - weighted Sobolev spaces KW - singular functions Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-34544 ER - TY - JOUR A1 - Ludwig, Julia A1 - Reymus, Marcel A1 - Winkler, Alexander A1 - Soliman, Sebastian A1 - Krug, Ralf A1 - Krastl, Gabriel T1 - Root maturation of an immature dens invaginatus despite unsuccessful revitalization procedure: a case report and recommendations for educational purposes JF - Dentistry Journal N2 - Background: The clinical management of teeth with complex dens invaginatus (DI) malformations and apical periodontitis may be challenging due to the lack of routine. The aim of this case report is to describe the endodontic treatment of an immature tooth with DI and to discuss strategies for preclinical training for teeth with such malformations. Case report: A 9-year-old male presented with an immature maxillary incisor with DI (Oehlers Type II) and apical periodontitis which was diagnosed by cone beam computed tomography (CBCT). Revitalization was initially attempted but then abandoned after failure to generate a stable blood clot. Nevertheless, considerable increase in both root length and thickness could be detected after medication with calcium hydroxide followed by root canal filling with MTA as an apical plug. Conclusions: The endodontic management of teeth with DI requires thorough treatment planning. In immature teeth, under certain conditions, root maturation may occur even with conventional apexification procedures. From an educational perspective, different strategies including CBCT and 3D-printed transparent tooth models for visualization of the complex internal morphology and redesigned 3D-printed replica with various degrees of difficulty for endodontic training, can be used to overcome the challenges associated with endodontic treatment of such teeth. KW - dens invaginatus KW - immature tooth KW - revitalization KW - apexification KW - 3D printing KW - dental education KW - replica Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-304108 SN - 2304-6767 VL - 11 IS - 2 ER -