Partial Differential Equations

Front Cover
Springer Science & Business Media, Dec 22, 1994 - Mathematics - 416 pages
This text is meant to be a self-contained, elementary introduction to Partial Differential Equations, assuming only advanced differential calculus and some basic LP theory. Although the basic equations treated in this book, given its scope, are linear, we have made an attempt to approach them from a nonlinear perspective. Chapter I is focused on the Cauchy-Kowaleski theorem. We discuss the notion of characteristic surfaces and use it to classify partial differential equations. The discussion grows out of equations of second order in two variables to equations of second order in N variables to p.d.e.'s of any order in N variables. In Chapters II and III we study the Laplace equation and connected elliptic theory. The existence of solutions for the Dirichlet problem is proven by the Perron method. This method clarifies the structure ofthe sub(super)harmonic functions and is closely related to the modern notion of viscosity solution. The elliptic theory is complemented by the Harnack and Liouville theorems, the simplest version of Schauder's estimates and basic LP -potential estimates. Then, in Chapter III, the Dirichlet and Neumann problems, as well as eigenvalue problems for the Laplacian, are cast in terms of integral equations. This requires some basic facts concerning double layer potentials and the notion of compact subsets of LP, which we present.
 

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Contents

II
1
III
2
IV
5
V
6
VI
8
VII
11
VIII
13
IX
14
CXXXVII
220
CXL
221
CXLI
223
CXLII
225
CXLIII
226
CXLIV
230
CXLV
231
CXLVI
232

XI
16
XIII
17
XV
19
XVI
20
XVII
22
XVIII
26
XIX
29
XXI
31
XXII
32
XXIII
33
XXIV
35
XXV
36
XXVII
38
XXIX
39
XXX
41
XXXI
42
XXXII
43
XXXIII
45
XXXIV
47
XXXV
48
XXXVI
49
XXXIX
50
XL
51
XLI
52
XLII
53
XLIV
54
XLV
55
XLVI
56
XLVII
57
XLVIII
60
XLIX
62
L
65
LI
66
LII
68
LIV
70
LV
72
LVI
76
LVII
77
LVIII
79
LIX
81
LX
85
LXI
88
LXII
89
LXIII
90
LXIV
92
LXV
93
LXVI
95
LXVII
96
LXVIII
98
LXIX
103
LXX
107
LXXI
108
LXXII
109
LXXIII
110
LXXIV
115
LXXVI
116
LXXVII
118
LXXVIII
120
LXXIX
123
LXXX
126
LXXXI
131
LXXXII
132
LXXXIII
134
LXXXIV
136
LXXXV
138
LXXXVII
139
LXXXVIII
140
LXXXIX
141
XC
144
XCI
146
XCII
150
XCIII
152
XCIV
154
XCVI
161
XCVII
162
XCVIII
163
C
165
CI
166
CII
167
CIII
168
CIV
169
CV
171
CVI
174
CVII
176
CIX
179
CX
181
CXI
182
CXII
183
CXIV
184
CXV
185
CXVI
186
CXVII
187
CXIX
188
CXX
191
CXXI
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CXXII
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CXXIII
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CXXIV
199
CXXV
200
CXXVI
201
CXXVII
203
CXXVIII
205
CXXIX
207
CXXX
209
CXXXII
213
CXXXIV
214
CXXXV
215
CXXXVI
219
CXLVIII
233
CL
236
CLI
237
CLII
238
CLIII
240
CLIV
244
CLV
245
CLVII
246
CLIX
249
CLX
251
CLXI
254
CLXIII
256
CLXIV
258
CLXVI
263
CLXVII
264
CLXIX
266
CLXX
267
CLXXI
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CLXXII
271
CLXXIII
274
CLXXIV
280
CLXXV
284
CLXXVI
286
CLXXIX
287
CLXXX
288
CLXXXII
291
CLXXXIII
292
CLXXXIV
293
CLXXXV
294
CLXXXVI
296
CLXXXVII
297
CLXXXVIII
298
CLXXXIX
300
CXC
301
CXCII
302
CXCIV
303
CXCV
304
CXCVII
308
CXCVIII
310
CXCIX
311
CC
312
CCI
313
CCII
314
CCIV
315
CCV
316
CCVI
318
CCVIII
319
CCIX
321
CCX
322
CCXI
325
CCXII
326
CCXIII
327
CCXV
328
CCXVI
330
CCXVII
335
CCXVIII
336
CCXIX
339
CCXX
340
CCXXI
341
CCXXIII
343
CCXXIV
344
CCXXV
345
CCXXVII
346
CCXXVIII
347
CCXXIX
348
CCXXX
349
CCXXXI
350
CCXXXII
351
CCXXXIII
352
CCXXXV
353
CCXXXVI
355
CCXXXVII
356
CCXXXIX
357
CCXL
359
CCXLI
362
CCXLII
364
CCXLIII
366
CCXLV
367
CCXLVI
368
CCXLVII
370
CCXLVIII
371
CCXLIX
372
CCL
375
CCLI
378
CCLII
379
CCLIII
380
CCLV
381
CCLVI
384
CCLVII
385
CCLVIII
386
CCLIX
388
CCLX
389
CCLXI
390
CCLXII
391
CCLXIII
393
CCLXIV
395
CCLXVI
396
CCLXVII
397
CCLXVIII
399
CCLXIX
401
CCLXX
403
CCLXXI
405
CCLXXII
406
CCLXXIII
407
CCLXXIV
409
CCLXXV
411
CCLXXVI
413
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