By Mariano Giaquinta
This monograph (in volumes) offers with non scalar variational difficulties bobbing up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as solid equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and obtainable to non experts. themes are handled so far as attainable in an ordinary means, illustrating effects with easy examples; in precept, chapters or even sections are readable independently of the final context, in order that elements may be simply used for graduate classes. Open questions are usually pointed out and the ultimate element of each one bankruptcy discusses references to the literature and infrequently supplementary effects. eventually, a close desk of Contents and an intensive Index are of aid to refer to this monograph
Read or Download Cartesian Currents in the Calculus of Variations I: Cartesian Currents (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics) PDF
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Discontinuous Groups of Isometries in the Hyperbolic Plane
This publication by way of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate historical past. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean aircraft, and this led him - in the course of
World battle II - to jot down the 1st chapters of the e-book (in German). while Fenchel,
who needed to break out from Denmark to Sweden due to the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what turned identified
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) used to be entire in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. even if, because of the speedy improvement of the topic, they felt
that colossal adjustments needed to be made earlier than book.
When Nielsen moved to Copenhagen college in 1951 (where he stayed until eventually
1955), he used to be a lot concerned with the foreign association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The documents of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and an entire manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The information additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes special reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen collage in 1956 (and stayed there
until 1974), used to be a great deal concerned with an intensive revision of the curriculum in al-
gebra and geometry, and centred his study within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally placed a lot attempt into his activity as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this task left him little time to complete the
Fenchel-Nielsen undertaking the best way he desired to.
After his retirement from the collage, Fenchel - assisted through Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the publication common Geometry in Hyperbolic house, which used to be released through
Walter de Gruyter in 1989 almost immediately after his demise. concurrently, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal the various imprecise issues that have been within the unique
manuscript. Fenchel informed me that he reflected removal components of the introductory
Chapter I within the manuscript, when you consider that this might be lined by way of the e-book pointed out above;
but to make the Fenchel-Nielsen e-book self-contained he finally selected to not do
so. He did choose to pass over
27, entitled Thefundamental crew.
As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed a superb activity of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus collage) with the facts studying of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a common dialogue of the variation to the fashion
of TEX. In such a lot respects we made up our minds to keep on with Fenchel's intentions. although, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of sure key-words, will be uniform through the booklet. additionally,
we have indicated the start and finish of an evidence within the ordinary form of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice reduction and delight they agreed to post the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this confident and quickly response. One
particular challenge with the book grew to become out to be the copy of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to bring those in ultimate digital shape, yet by way of 1997 it grew to become transparent that he
would no longer be capable to locate the time to take action. notwithstanding, the writer provided an answer
whereby I may still carry distinctive drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they'd manage the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his positive
collaboration in regards to the real construction of the figures.
My colleague Bent Fuglede, who has personaHy identified either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right viewpoint. In
this connection i need to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this ebook reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few targeted symbols, a listing of notation with brief
explanations and connection with the particular definition within the e-book has been incorporated. additionally,
a complete index has been further. In either circumstances, all references are to sections,
not pages.
We thought of including an entire record of references, yet made up our minds opposed to it as a result of
the overwhelming variety of examine papers during this sector. as an alternative, a miles shorter
list of monographs and different accomplished money owed suitable to the topic has been
collected.
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this book into lifestyles.
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Extra resources for Cartesian Currents in the Calculus of Variations I: Cartesian Currents (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)
Example text
2. Recall that f : X R is a Borel function if the inverse image of an open set is a Borel set. Also recall that not every Lebesgue measurable function f : Rn -, JR can be obtained as a monotone limit of continuous functions, but we have Theorem 3 (Vitali-Caratheodory). e. pointwise limit of a non-decreasing sequence of functions cpk which are upper semicontinuous and bounded from above. If moreover f is integrable, we have lim f (f - Wok) dx = 0 . Finally next theorem shows an important relation between measurable and continuous function.
Ak) is a vector valued measure if and only if each µk is a signed measure. We also have Proposition 2. The total variation I vl of a vector valued measure v can by equivalently defined as I vI (A) = sup{E00I v(Ak) I I Ak E B ,(X) , Ak disjoint , UkAk C A} , k=1 for A E 15,(X). 4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems 27 Let v = (v' , ... , v') and µ be respectively a vector valued measure and a Radon measure on X. We say that v is absolutely continuous with respect to p, V << µ, if lvl << µ .
Consider the measures 1-tk and 7{1, k _< n, in R n. Although they can be very different, it is easily seen that 7dk (A) = 0 if and only if Hk 00 (A) = 0. e. x E E, and one can show that, if Q, Qv v = 1, 2.... are compact se and if every open set A D Q contains Q for v sufficiently large, then 7{k (Q) > lim sup 7{1 (Qv) . v-oo Notice that the last inequality is in general false for the Hausdorff measure 7-Ck. The properties of 7l"0 illustrated above sometimes make the use of 7-lOO more convenient than the use of 7{k.