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Carleman-Sobolev classes and Green’s potentials for weighted Laplacians
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is based on two papers: the first one concerns Carleman-Sobolev classes for small exponents and the other solves Poisson's equation for the standard weighted Laplacian in the unit disc.

In the first paper we start by noting that for small Lp-exponents, i.e. 0<p<1, the way we usually define Sobolev spaces is very unsatisfactory, which was illustrated by Peetre in 1975. In an attempt to remedy this we introduce completions of a class of smooth functions, which we call Carleman-Sobolev classes since they generalize Sobolev spaces and uses a norm inspired by Carleman classes. If the class is restricted with a growth condition on the supremum norms of the derivatives, we prove that there exists a condition on the weight sequence in the norm which guarantees that the resulting completion can be embedded into C(R). This condition is even sharp up to some regularity on the weight sequence, in the sense that the norm inequality required for continuity no longer holds. We also show that the growth condition is necessary, in the sense that if we drop it entirely we can naturally embed Lp into this class's completion. Hence in this case we cannot consider the completion as a proper generalization of a Sobolev space.

In the second paper we find Green's function for the standard weighted Laplacian and give conditions on the Riesz-mass such that we can use Green's potential to solve Poisson's equation with zero boundary values in the sense of radial L1-means. The weight here comes from the theory of weighted Bergman spaces and from this context it gets the label as the standard weight.

Abstract [sv]

Den här avhandlingen är baserad på två artiklar: den första handlar om Carleman-Sobolev-klasser för små exponenter och den andra löser Poissons ekvation för den standardviktade Laplacianen i enhetsskivan.

I den första artikeln börjar vi med att notera att för små Lp-exponenter, dvs 0<p<1, så är metoden man vanligen använder för att definiera Sobolevrum väldigt otillfredsställande, detta illustrerades av Peetre i en artikel från 1975. I ett försök att förbättra situationen introducerar vi tillslutningar av klasser av släta funktioner, som vi kallar Carleman-Sobolev-klasser eftersom de generaliserar Sobolevrum och använder en norm inspirerad av Carlemanklasser. Om klassen inskränks med ett växtkrav på derivatornas supremumnormer, så visar vi att det finns ett krav på viktsekvensen i normen som garanterar att den resulterande tillslutningen går att inbädda i C(R). Detta krav är dessutom skarpt upp till viss regularitet hos viktsekvensen, i meningen att normolikheten som krävs för kontinuitet inte längre håller. Vi visar också att växtkravet är nödvändigt, i meningen att om vi utelämnar detta krav så kan vi inbädda Lp naturligt i tillslutningen av denna klass. Alltså, utan kravet kan vi inte betrakta tillslutningen som en äkta generalisering av ett Sobolevrum.

I den andra artikeln hittar vi Greens funktion för den standardviktade Laplacianen och ger krav på Rieszmassan som tillåter oss att använda Greens potential för att lösa Poissons ekvation med noll på randen i betydelsen av radiella L1-medelvärden. Vikten kommer från teorin om viktade Bergmanrum och det är där den kallas för standardvikten.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , vii, 18 p.
Series
TRITA-MAT-A, 2014:06
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-145012ISBN: 978-91-7595-134-8 (print)OAI: oai:DiVA.org:kth-145012DiVA: diva2:715649
Presentation
2014-05-20, 3418, Lindstedtsvägen 25, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2012-3122
Note

QC 20140509

Available from: 2014-05-09 Created: 2014-05-05 Last updated: 2014-05-09Bibliographically approved
List of papers
1. Carleman-Sobolev classes for small exponents
Open this publication in new window or tab >>Carleman-Sobolev classes for small exponents
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper is devoted to the study of a generalization of Sobolev spaces for small Lp exponents, i.e. 0<p<1. We consider spaces defined as abstract completions of certain classes of smooth functions with respect to weighted quasi-norms, simultaneously inspired by Carleman classes and classical Sobolev spaces. If the class is restricted with a growth condition on the supremum norms of the derivatives, we prove that there exists a condition which guarantees that the resulting space can be embedded into C(ℝ). This is sharp up to some regularity conditions on the weight sequence. We also show that the growth condition is necessary, in the sense that if we drop it entirely we can naturally embed Lp into the resulting completion.

Keyword
Generalized Sobolev spaces, Denjoy-Carleman classes, spline approximation, small exponents.
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-145010 (URN)
Funder
Swedish Research Council, 2012-3122
Note

QS 2014

Available from: 2014-05-05 Created: 2014-05-05 Last updated: 2014-05-09Bibliographically approved
2. Solving Poisson's equation for the standard weighted Laplacian in the unit disc
Open this publication in new window or tab >>Solving Poisson's equation for the standard weighted Laplacian in the unit disc
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We will find Green's function for the standard weighted Laplacian and use the corresponding Green's potential to solve Poisson's equation in the unit disc with zero boundary values, in the sense of radial -means, for complex Borel measures  satisfying the condition  for .

Keyword
Green’s function, standard weighted Laplace operator, Poisson’s equation.
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-145007 (URN)
Note

QS 2014

Available from: 2014-05-05 Created: 2014-05-05 Last updated: 2014-05-09Bibliographically approved

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