Partial Differential Relations

Front Cover
Springer Science & Business Media, 1986 - Mathematics - 363 pages
The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space). Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field. The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book.
 

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Contents

Section 1
1
Section 2
25
Section 3
28
Section 4
48
Section 5
64
Section 6
101
Section 7
115
Section 8
128
Section 17
198
Section 18
202
Section 19
207
Section 20
221
Section 21
224
Section 22
227
Section 23
233
Section 24
235

Section 9
135
Section 10
151
Section 11
160
Section 12
168
Section 13
171
Section 14
174
Section 15
177
Section 16
189
Section 25
258
Section 26
277
Section 27
280
Section 28
284
Section 29
288
Section 30
350
Section 31
359
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