ERRATA
Wherever the expression B(R, X) appears (or any similar expression with the
radius listed before the center), it should be replaced by B(X, R) (with the center
listed first). Also, “Poincarè” should read “Poincaré” wherever it appears.
page v
line -13 “completely” should read “completed”.
page 5
line 3 “basis” should read “basic”
line -13 Ω is always a bounded domain.
page 8
line 4 The inequality Lu < 0 should read Lu > 0.
page 10
line -2 The exponent at the end of the line should be α(r2 − R2 ).
line -1 should read
2
≥ e−αR [α2 λR2 − α(2Λ + 2Λ1 R + η 2 ) − Λ2 ].
page 12
line 1 Delete the factor of 3/2 in the definition of R0 and replace y0 with
y1 .
line 2 The inequality for x should read |x − y| < 32 R0 .
line 3 The final inequality should read R02 ≤ η 2 (s − t) ≤ 0.
line 4 Replace y0 with y1 .
line 6 Replace x0 with x1 .
line 7 Replace “minimum” with “maximum”.
line 10 Replace s = t0 with s < t0 .
In the hypotheses of Lemma 2.8, add the condition 16Λ1 R ≤ 1 and change
the assumption on Λ2 to 4Λ2 R2 ≤ 1.
In condition (2.9), replace 3 with 4.
line -9 Replace 3 by 4 in the definition of v.
line -8 Replace the inequality 0 ≤ v ≤ 3Λ0 with 0 ≤ v ≤ 4Λ0 R2 .
line -7 should read
Lv = −2T − 2bi (xi − y i ) + cv + 4Λ0 ≥ Λ0 (2 − (16Λ1 R)1/2 − 4Λ2 R2 ) ≥ 0.
page 13
page 16
page 17
line -6 The coefficient in front of v should be ε/(4Λ0 R2 ).
line 11 Replace R2 with η 2 .
line 7, “Theorem 2.6” should read “Theorem 2.11”.
line -7 “(2.4b,c)” should read “(2.5b,c)”.
line 4 Replace t0 by t1 .
Equation (2.22) should read
Q = {|x − x0 | < R} × (t1 , t1 + hR2 )
page 18
page 25
page 32
line -8 [130] should read [131].
The z’s in the integral on the left side of (3.5) should be x’s.
Lemma 3.15 should include the hypotheses η ∈ [0, 1] and R ≤ λ/2 (instead
of R ≤ λ).
The last inequality in condition (3.16) should read
|f | + |Df | + |ft | ≤ F.
page 33
page 35
line -6 Replace ηt by 2ηt, ηR by 2ηR, and R by 2R.
line -1 Replace k1 + F by k1 + Φ + F .
In Theorem 3.16, change R ≤ λ to R ≤ λ/2.
In Theorem 3.18, change R ≤ λ to R ≤ λ/2.
1
2
page 36
page 37
page 38
page 39
page 40
page 41
page 43
page 46
page 47
page 49
page 50
page 65
In (3.21), replace |Df |2 + |ft |2 by |f |2 + |Df |2 + |ft |2 .
In (3.21)0 , replace |Df |2 + |ft |2 by |f |2 + |Df |2 + |ft |2 .
In line -14, F should also satisfy sup |F | ≤ sup |f |.
In line -12, replace |DF |2 + |Ft |2 by |F |2 + |DF |2 + |Ft |2 .
In lines -11 and -8, replace |Df |2 + |ft |2 by |f |2 + |Df |2 + |ft |2 .
In lines 7, 17, and -10, replace R ≤ λ by R ≤ λ/2.
In line 10, replace R ≤ λ by R ≤ λ/2.
In lines 11 and -2, replace R ≤ λ by R ≤ λ/2.
The second inequality in line -9 should read |X − X0 | ≥ δ, rather than
|X − X0 | < δ.
In line -13, replace R ≤ λ by R ≤ λ/2.
In line 10, replace R ≤ λ by R ≤ λ/2.
Line -9, replace “points” by “point”.
Line -4 The correct definition of w1 is as follows: First, let w0 be a nonnegative, continuous function in Rn+1 , which agrees with w on ω(t0 ) and
which vanishes only at x0 . (Such a function exists by Exercise 3.1.) Then
w1 is the solution of Hw1 = 0 in Q and w1 = w0 on PQ.
In Exercise 3.11, the definition of a tusk should read
2
{X : t0 − T < t < t0 , [x − x0 ] − (t0 − t)1/2 x1 < R2 (t0 − t)}
line -2 “Corollary 4.11” should read “Corollary 4.10”
(b)
In the definition of hf ia , the supremum should be taken over all X 6= Y
in Ω with x = y.
line -13, Replace εa with ε1−a .
line -12, Replace (Ub /Ua )(1−a)(b−a) with (Ua /Ub )1/(b−a) .
q > n + 2 should read q > n + 4.
In the proof of Lemma 4.16, the correct inequality for Lw1 is Lw1 ≤ 4(α −
1)αλdα−2
and the correct definition for w is
1
Ψ
2F
F
+
w2 .
w1 +
w=
4(1 − α)αλ 2αc(µ, µ̄)
α
page 52
page 53
page 57
line 15 (2.28) should read (2.24)
line -10 (2.28) should read (2.24)
line 13 The left side of this inequality should read
Z
|Dw|2 dX
page 59
page 66
line 9 “coeficients” should read “coefficients”
After Lemma 4.16, note that c(µ, µ̄) = µ(1 − µ̄2 )1/2 − µ̄, so the constant C
in (4.40) depends only on a lower bound for µ̄.
line 7 “Theorem 4.14” should read “Theorem 4.15”
line -10 “Morover” should “Moreover”
line -1 (4.72,) should read (4.72)
line 20 (4.22b) should read (4.20b)
line -13 after “construction”, add “; see also [220]”.
line -13 “cite 175” should read [175].
line -8 Before “Earlier”, add “Our study of the oblique derivative problem,
starting on page 65, comes from [174].”
line -9 “[240]” should read “[241]”.
page
page
page
page
71
80
81
82
page 83
3
page 84
page 89
page 92
page 95
page 96
page 97
page 102
page 104
page 108
page 112
page 114
In Exercise 4.1, “Corollary 4.11” should read “Corollary 4.10”.
line -7 R2 should read R2 .
line 17 “ continous” should read “continuous”
line -15 “respectivelyl” should read “respectively”
line 9 “Theorem 5.11” should read “Theorem 5.10”
line 5 “Theorem 5.16” should read “Theorem 5.15”
line -14 (6.1c) should read (6.2c)
line 11 “readly” should read “ready”
line -9 “supppose” should read “suppose”
line -2 “Corollary 6.7” should read “Corollary 6.10”
Lemma 6.13 should read “Let Ω be a domain in Rn , let ηRbe a nonnegative
continuous function in Ω, and supposeR f ∈ L∞ (Ω) with Ω f η dx = 1 and
sup |f | ≥ 1. If u ∈ Lp (Ω) and if uf = Ω uf η dx, then
Z
|u − uf |p η dx ≤ 2p sup |f | inf
Z
L∈R
Ω
|u − L|p η dx.”
Ω
line -8 should read
Z
p
1/p
|u − uf | η dx
page
page
page
page
120
121
122
126
page 127
p
≤
Ω
page 115
1/p Z
1/p
p
|u − L| η dx
+
|L − uf | η dx
Z
Ω
Ω
In Proposition 6.14, the ball B has radius R, the factor |B| should not be
present in (6.18), and Rn+p should read Rp in line 8.
line -6 X ∈ (2R) should read X ∈ Q(2R)
line -10 M should read kR
line 2 Q(R, Y ) should read Q(4R, Y )
line 9 “becasue” should read “because”
line -8 delete the parenthesis.
line -5 “betwen” should read “between”
line 2 should read
!
Z
η 2 ln ū dx ,
K = exp
B(3R)×{s−4R2 }
page 128
page
page
page
page
page
page
137
138
140
145
147
148
page 149
page 150
page 151
The comma at the beginning of line 5 should be at the end of line 4.
line 12 “prinicple” should read “principle”
line -4 (6.1) should read (6.2).
line -3 “Lemma 6.37” should read “Lemma 6.36”.
line 7 “Theorem 5.17” should read “Theorem 5.18”.
line -6 “costant” should read “constant”
line 13 b should read bξ 2 .
line 1 Q+ should read Q.
line -8 “Lemma 6.52” should read “Lemma 6.48”.
line -4 “Lemma 6.53” should read “Proposition 6.51”.
line 7 “Proposition 6.52” should read “Lemma 6.48”.
line 7 “[281]” should read “[282]”.
line -15 Add the sentence “Lemma 6.13 is based on [18, Lemma 2].”
line 10 “[280]” should read “[281]”.
4
page 152
In Exercise 6.2, the definition of kukp,q should read

!q/p 1/q
Z
Z
p
kukp,q = 
dt .
|u(X)| dx
I(Ω)
page 153
page 156
Ω(t)
In exercise 6.4(b), limp→−∞ should read limp→−∞ kukp .
Equation (7.2a) should read
B0 = kb/D∗ kn+1
n+1 + R + β0 .
n/(n+1)
page 157
page 159
page 161
In (7.2b), replace B0 (R + β0 )n/(n+1) by B0
line 12 should read
ωn M n+1
Φ(E + (u)) ≥ |Σ| =
.
n(n + 1)2n+1 (R + β0 )n
−1/(n+1)
line -4, define µ = B0R
kf − /D∗ k.
line -3 Replace G by Σ g(ξ) dΞ.
line 7 the integrand in the second integral should read
Z
f (X) − 1
f (X + Y + ρm (X)) dY |Km |
Km
page 169
page 176
page 179
page 180
page 181
line 2 “Theorem 6.3” should read “Theorem 6.6”.
line -15 “Theorem 5.16” should read “Theorem 5.15”.
line -10 “modificiation” should read “modification”.
line -7 “Proposition 7.10” should read “Proposition 7.14”
line 7 “dxn ” should read “dX”.
Condition (7.37a)0 should be included in condition (7.37a).
In the statement of Theorem 7.21, C is determined also by ρ and λ1 .
2
In the proof of Theorem 7.21, ζ should be defined as ζ = (1−|x| /R2 )+ (1+
2 +
t/R ) .
The definition of kukp,q should read
!1/p
Z
kukp,q =
sup
r≤diam Ω
p
r−q
|u| dX
.
Q(r)
Condition (7.37b)0 should read
kb/λkn+1,1+θ ≤ B
page 182
for some θ ∈ (0, n + 1), and C and p are determined only by BRθ/(n+1) , n,
λ0 , Λ0 , µ, and θ in Theorem 7.22.
The exponent of R on the right side of inequality (7.39) should be −n − 2.
In the statement of Lemma 7.23, the inequality in (7.40) should be reversed,
the inequality Brn/(n+1) ≤ β0 should read Brθ/(n+1) ≤ β0 , and C1 depends
also on θ but not on B.
In the proof of Lemma 7.23, the definitions of v and Q∗ should have R
replaced by r.
In the proof of Lemma 7.23, the expression of Lv should have R replaced
by r and Q(R/2) should read Q(r/2). Also, the estimate on Lv should read
rn/(n+1) kLv/D∗ kn+1,Q∗ ≤ Ck + C[ζ 1/(n+1) + β0 ]h.
5
In Lemma 7.24, β1 is independent of λ0 , the constants depend also on θ,
and the inequality Brn(n+1) ≤ β1 should read Brθ/(n+1) ≤ β1 .
In (7.43), replace s with x.
In fact, ψ is only C 2,1 on the subset of Rn+1 on which t > −(1 + (α −
1)ε2 )/(1 − ε2 ), which is true for X ∈ Q.
line -1 should read
L0 ψ = ψ −q [8aij xi xj − ψ1 4T +
page 183
(1 − ε2 )q 2
1 − ε2
ψ1 + 2
ψ1 ],
αψ0
α
Replace Λ0 by λ1 everywhere.
The estimate for Lψ should read
Lψ ≥ bi Di ψ = −4ψ1 ψ0−q b · x ≥ −4|b|r(εr)2−2q .
Condition (7.44b) should read
2
ψ(x, (α − 1)r2 ) = (r2 − |x| )2 r−2q ≥
9r−2q+4
16
if |x| ≤ r/2.
In the last paragraph of the proof of Lemma 7.24, the second to last sentence
should read “Applying Theorem 7.1 to v then yields v ≥ −Ck − Cβ1 Aε−2
in Q and hence
ε2q−4
− Ck
2
for |x| ≤ r/2 and t = (α − 1)r2 by (7.44b) provided β1 is small enough.”
The reference to (7.53c) should be (7.45c).
line 2 “Rn/(n+1) ” should read “Rθ/(n+1) ”.
In the definition of m, the second inequality should read h > γ (m−1)q h1 .
line -11 “Rn/(n+1) ” should read “Rθ/(n+1) ”.
In the statement of Theorem 7.29, condition (7.37) should be condition
(7.37a).
In inequality (7.51), Ω(r) should be Ω[r].
line -3 “B” should read “BRθ/(n+1) ”, and C and α depend also on θ.
lines 4 and 5 “r” should read “R”.
In estimate (7.53), the constant 8 should be 8/δ.
line -7 “t = −r2 ” should read t = −4r2 ”.
The second to last sentence in the proof of Lemma 7.31 should read “Since
w1 ≤ 3xn in G(ρ, r), it follows that u ≥ [A/4 − (2/δ)F1 (ρr)δ ]xn in G(ρ, r).”
line 13 “Lemma 7.5” should read “Lemma 7.4”.
line 20 “Theorem 7.19” should read “Lemma 7.19”.
line 17 “Lemma 7.33” should read “Lemma 7.32”.
line -18 “Apushkinaskaya” should read “Apushkinskaya”.
line -4 “Lemma 7.33” should read “Lemma 7.32”.
line -6 “solvabilility” should read “solvability”.
line 17 “Lemma 4.2” should read “Proposition 4.2”.
line 18 “verfication” should read “verification”.
line 5 “efffect” should read “effect”.
line 5 add the sentence “Another approach, similar to that described in
Exercise 5.4, was laid out in [132, 133] for solutions of the Cauchy problem.”
line -19 “[L11]” should read “[178]”.
u ≥ A[(εr)2q−4 ψ − Cβ1 /ε2 ] − Ck ≥ A
page 184
page 185
page 186
page 187
page 188
page 193
page 194
page
page
page
page
page
195
202
206
209
211
6
page 215
page 218
page 219
page 222
line -7 “is is” should read “is in”.
line 9 “modifed” should read “modified”.
line 1 “(9.5a,b)” should read “(9.6a,b)”.
line 4 “(9.5a)0 ” should read “(9.6a)0 ”.
line 10 “detemined” should read “determined”.
The conclusion of Exercise 9.3 should read
sup u ≤ max{M, sup u+ } + 2LR.
Ω
page 223
page 228
page 229
line
line
line
line
-6
-2
-5
-9
∂Ω
“(9.5a)” should read “(9.6a)”.
“auxillary” should read “auxiliary”.
“it exterior Hθ condition” should read “exterior Hθ condition”.
the last term should read
Cµ(f 0 )ζ−2 dζ−2 E
page
page
page
page
230
231
238
240
page 246
page 247
page 248
page 249
page 256
page 259
page 267
page 268
page 269
page 270
page 271
line
line
line
line
line
line
line
line
line
line
line
line
line
line
line
-7 “convexs” should read “convex”.
5 after “(10.8)” insert the word “holds”.
13 “folows” should read “follows”.
5 “complieted” should read “completed”.
-9 “chocies” should read “choices”.
-12 “uniformlly” should read “uniformly”.
4 P should read PΩ.
-11 “continous” should read “continuous”.
-7 “Theorems” should read “Theorem”.
14 [252] should read [250].
-8 “same is true” should read “same is true for ω ∗ ”.
-20 [93] should read [83].
13 [165] should read [166].
-11 “replce” should read “replace”.
-13 should read
−1/2
1
2|A∞ |
+
4σ ≤
(K θ σ)2
(K θ σ)2
line -7 “Corollary 6.7” should read “Corollary 6.10”.
line 3 “ilustrate” should read “illustrate”.
line -9 “Theorem 11.4” should read “Theorem 11.1”.
In line 6, Aiz should be multiplied by pk .
In (11.44), there is a missing factor v in the term |ζζt | and a missing factor
of ζ 2 in the term β12 Λ0 .
line 7 The factor v multiplying E1 should be v[χ + (v − τ )+ χ0 ].
lines 7, 8, 9 The integrands in these four integrals are all missing a factor
of ζ 2 .
line 10 There should be an additional term of
Z
v 2 |ζζt | χ dX.
Ω(s)
2
page 272
line -9 The term in square brackets should read 9µ̄ |Dζ| v + |ζζt | v 2 .
In condition (11.45b), “increasing” should read “decreasing”
The integrals in (11.48) and (11.49) should both be multiplied by an additional factor of ρ−n−2 , and the exponents −N − 2 should read N + 2.
7
Add to the hypotheses of Lemma 11.11
λ≤Λ
page 273
line 2 Delete the first factor of v in the integrand.
line -9 should read
|δh|2 ≤ C(n, β)qχ[E1 ζ (N +2)q−N + ρ−2 λvζ (N +2)(q−1) ]
page
page
page
page
page
274
279
280
281
282
page 283
page 284
line -6, w2q should read wq .
line 11 “managable” should read “manageable”.
line 11 “Lemma 4.25” should read “Lemma 4.24”.
line -5 α + 2 should read α + 1.
line 3 “kown” should read “known”.
line -3 b0 should read b1 .
line -13 “Athough” should read “Although”.
In (11.62) and in line 9, the factor of 2 can be removed.
In (11.66), the factor of 2 can be removed.
Lines 9 and 11, the factor of 2 can be removed.
inequality (11.67) should read
|ux | ≤ exp(4β0 M )(L0 + β1 + 4M/r)
page 285
page 286
page 289
page 291
page 294
page 295
the term β2 t[x − y] should not be present in the definition of v ± .
line 2 The correct expression for b1 is β1 + 4M/r.
Equation (11.69) only holds in Q(r).
line -10 “Section 4” should read “Section 5”.
i
i
line -11 Ckij should read cij
k , b should read b̄ .
ijr
ijm
i
line -8 a should read ā , b should read b̄i , there is a missing term of
k
cij
k Dij uν inside the square brackets.
line 1 after the first sentence, add “In addition, Krylov [140, 141] has developed an alternative method for proving gradient bounds, which also applies
to a large class of degenerate, fully nonlinear equations.”
In Exercise 11.7, “Lemma 11.16” should read “Lemma 11.15”.
lines 17 and -4 dist(Ω, PΩ) should read dist(Ω0 , PΩ)
line -7 should read
−wt± + Di (aij Dj w± + fki ) + aijm Dm w±
page 300
line -5 The expression for fki should read a[δki + 2εDi u].
In inequality (12.14), “F0 ” should read “F0 + [Dϕ]β ”, and the proof of
Lemma 12.6 should be modified as follows: First, in the definition of Q̃(ρ),
R2 should read ρ2 . Then take
M (R)
+ [Dϕ]β (w2 )(1+β)/2 .
w = 2F0 w1 +
R1+β
It follows that w ≥ ±v in Q̃(R), so
1+β r
M (R)
M (r) ≤ 2F0
+
+ [Dϕ]β 2r1+β .
R1+β
(1 + 2Λ)1/2
page 301
From this inequality, the proof is completed as written.
n β−1
line 5, t(β−1)/2,λ[x ]
should read t(β−1)/2 , λ[xn ]β−1
8
In condition (12.14)0 , F0 should read F0 + [Dϕ]β + hϕi(1+β)/2 , and the proof
of Lemma 12.8 should be modified by setting F1 = [Dϕ]β + hϕi(1+β)/2 and
taking
M (R)
(1+β)/2
w = 2F0 w3 +
+
F
.
1 (w2 )
R1+β
Then we have
M (R)
M (r) ≤ 2 F0 1+β + F1 r1+β ,
R
page
page
page
page
303
306
307
308
page 309
page 312
page 316
which implies (12.14)0 .
line 3 “inequalites” should read “inequalities”.
The section number is 6, not 5.
line 11 “replaced” should read “relaxed”.
line -14 “Dirchlet” should read “Dirichlet”.
line -3 add after “...x and z” the phrase “with zH(X, z) ≤ kz 2 + b1 for
some nonnegative constants k and b1 ”.
line -1 replace both instances of Du with p.
line 9 “Corollary 7.32” should read “Lemma 7.32”.
line -5 “[53, 178]” should read “[53, 178, 247]”.
line 3 “fuction” should read “function”.
condition (13.3a) should read
sgn zb(X, z, − sgn zM1 γ) < 0
page 320
page 322
page 327
line -4 “abritrary” should read “arbitrary”.
line 14 “auxillary” should read “auxiliary”.
In (13.22c), Λ0 should be replaced by Λ0 (|p|/C(c0 )), where
1/2
(1 − β2 )2
,
c0 = 1 −
1 − (1 − β2 )2 + (β2 + β3 )2
and C(c0 ) is the constant from (13.15).
In (13.26c), the power of |p| should be one.
In (13.26d), the left side of the inequality should be ε1 Cki pi ξ k .
An additional hypothesis is needed here:
(13.27d)
ε1 v p · Az + aii + B ≤ β42 Λ0 .
page 328
page 329
page 330
page 331
page 332
In inequality (13.31), the second integral is missing a dt.
In the first equation after (13.31), the term v −1 Aij Djk Di ua0 should read
v −1 Aij Djk uDi uak0 , and the term ak0 Cki Di u should read ak0 Cki Di uθ.
In line 4, (13,22a) should read (13.22a).
The second and third displays should be interchanged, and a0 should read
ak0 .
The last two integrals on this page are missing a factor of v1 in the integrands.
On lines 11 and 12, the second and third integrals are missing a dt.
In the estimate of η b̄b̄z ut (1 − vτ1 )χζ 2 , we must invoke (13.27e).
The estimate for the term involving b̄t should read
τ
η b̄b̄t (1 − )χζ 2 ≤ Cβ42 Λ0 χζ 2 v.
v1
In (13.35c), ξi ξj should read ζik ζjk .
9
page 333
In (13.39), the second integral should be over Ω(τ, 2ρ).
On lines -4, -2, and -1, replace dσ by ds.
Line 2 delete the second “we”.
Line 4 should read
Z τ 2
I0 (t) ≤ KEΨσ
wq−2 w0 + wq−1
|δv1 |ζ q dµ
v1
In lines 6 and -9, the factor of τ in front of the integrals should be deleted.
Lines 12 and 13 should read
Z
J1 ≤ KEβ5 σ (wq−2 λ0 |δv1 |2 /(v1 )2 )1/2 (wq ν · A)1/2 dµ
Z
≤ KEβ5 σ( wq−2 E2 ζ q dµ)1/2 (J 0 )1/2 .
page 334
page 335
page 336
Line 1 “Suppse” should read “Suppose”.
In example (i), we need ψz ≡ 0 and ε1 ≡ 0 (so Ψ1 = 0).
line 10 The set of integration should be Ω ∩ {|x − x0 | < 2ρ, v1 ≥ τ }.
The correct structure functions in example (ii) are
Λ0 = Kv13 , Λ = Kv12 , λ ≡ min{1, θ0 }.
page 341
line 9 (13.23) should read (13.33).
Line -15 “intial” should read “initial”.
Condition (13.49) should read
±b(X, z, ±β0 γ(X)) > 0
0
for X ∈ Q (R) and |z| < M0 .
Theorem 13.16 needs two additional hypotheses. First,
|A(x, t, z, p) − A(x, s, z, p)| + |ψ(x, t, z) − ψ(x, s, z)| ≤ µK |t − s|1/2
page 342
page 343
page 348
page 349
page
page
page
page
351
352
353
361
page 362
page 365
page 366
page 367
page 369
page 372
for all (x, z, p) ∈ ∂ω × R × Rn with |z| + |p| ≤ K and all s and t in (0, T ).
Second, ϕ ∈ H2 . Consequently, C in the conclusion also depends on |ϕ|2 .
line -11 Σ0 (, R) should read Σ0 (R).
line -2 “qunatities” should read “quantities”.
In estimate (13.58) and in the following line, Dv should read Du.
line -12 The word “continuous” is misspelled.
lines 6 and 7: Replace Q+ and Q0 by Σ+ and Σ0 .
line 8 Replace aij (X0 , Du) by aij (X0 , Dv).
line 12 “(13,44a,b)” should read “(13.44a,b)”.
line 10 add reference [263] to the list.
line 2 [185] should read [186]. line 14 “satsifies” should read “satisfies”.
line 14 “auxillary” should read “auxiliary”.
line-4 “L∗ ” should read “Λ∗ ”.
line -13 “satisfiy” should read “satisfy”.
line 11 “auxillary” should read “auxiliary”.
line -6 The reference to equation (14.17) should be to (14.15).
line 11 b − 2 should read b2 .
line 11 “eignevalues” should read “eigenvalues”.
line -15 “abitary” should read “arbitrary”.
line -8 “Lemma 11.3” should read “Lemma 11.4”.
lines 1, 2, 7 Replace δ by θ.
10
page
page
page
page
375
377
379
386
page 389
line 8 “Theorem 14.2” should read “Corollary 14.2”.
line 12 a
n u should read aν .ij line 17 aij
ν should read aν,p .
line -14 “weaking” should read “weakening”.
line -1 “14.11” should read “14.12”.
line 10 “Lemma 7.35” should read “Lemma 7.32”.
line 16 “Theorems 14.17” should read “Lemma 14.17”.
line 8 This sentence should read “To state them, we use κ = (κ0 , . . . , κn−1 )
to denote the space-time curvature of ∂Ω as defined in Section 10.3.”
line -12 “nonnnegative” should read “nonnegative”.
line 7 The expression in square brackets should read 12 |x|2 − t.
line -11 should read
X
L(u − u) ≤ c1 [u − u] − ε1 [Fτ +
F ii ] − δ0 + σ(ε1 ).
i
page 400
page 402
page
page
page
page
page
404
405
409
414
415
page 416
page 420
page 421
page 422
page 423
page 424
page 425
page 427
line 4 “roolts” should read “roots”.
line 1 The range for j should be 1 ≤ j < k.
line 2 S0,i should read S1,k .
line 3 j = 0 should read j = 1.
(15.b) should read (15.4b).
line 15 “corresponding” should read “corresponding”.
line 12 “calcuation” should read “calculation”.
line 17 “esential” should read “essential”.
line 11 “ Exercise 15.6” should read “Exercise 15.5”.
line 17 after [271], add “and Wang and Wang [273]”.
line -15 “[28]” should read “[30]”.
line -13 “form” should read “from”
Reference [22] has appeared in volume 125 (1997), 1785–1792.
In Reference [31] “Calderòn” should read “Calderón”.
In Reference [32] “proerties” should read “properties” and “Calderon” should
read “Calderón”.
In Reference [39] “de” should read “di”.
In Reference [39] “essistenza” should read “esistenza”.
The page numbers for Reference [61] are 458–520.
In Reference [64] “oblique” should read “obliqua”.
In Reference [64] “ellitiche” should read “ellittiche”.
In Reference [65] “dalle” should read “della”.
In Reference [65] “oblique” should read “obliqua”.
In Reference [66] “oblique” should read “obliqua”.
In Reference [66] “ellitiche” should read “ellittiche”.
In Reference [77] “divergnece” should read “divergence”.
In Reference [79], “Prolbems” should read “Problems”.
In Reference [95], “ssolvability” should read “solvability”.
In Reference [98], the page numbers for the Russian article are 193–206.
The source for Reference [105] is: in Geometric Analysis and the Calculus
of Variations, International Press, Cambridge, MA (1996), 125–141.
In Reference [136] add “(Russian)” after 487–523.
In Reference [138] add “(Russian)” after 23–26.
In Reference [139] replace “yr” with a comma.
11
page 428
page 432
page 434
In Reference [152] add “(Russian)” after 161–240.
In Reference [217] add “(Russian)” after 37–46.
In Reference [239] “sufrace” should read “surface”.
In Reference [240] add “3–163 (Russian); English transl in Proc. Math.
Inst. Steklov 83 (1966), 1–184”.
page 436
In Reference [280], “Q-growth” should read “Q-minima”.
Reference [281] has appeared in volume 10 (1997), 31–44.
Reference [283] has appeared in volume 348 (1996), 2811–2844.
Also add the following references:
References
[OK]
[La]
O. A. Oleinik and S. N. Kruzhkov, Quasilinear second-order parabolic equations with many
independent variables, Usp. Mat. Nauk 16 (1961), no. 5, 115–156 [Russian]; English transl.
in Russian Math. Surveys 16 (1961), no. 5, 105–146.
E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, Nauka, Moscow,
1971 [Russian]; English transl. Amer. Math. Soc., Providence, R.I., 1998.