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## Full text

(1)

### Edger Sterjo

The Graduate Center, City University of New York

### How does access to this work benefit you? Let us know!

This work is made publicly available by the City University of New York (CUNY).

(2)

(3)

(4)

(5)

(6)

(7)

(8)

d

(9)

p−2

∂Ω

d

2

d−22d

01,2

p

2

(10)

2d−2d−2

∂Ω

0

2

(11)

p−2

∂Ω

0

0

01

p−2

∂Ω

1

θ

0

∂θ

θ

θ

(12)

Ku2

∂Ω

01

t2

t→∞

βt2

|t|α

p−2

(13)

l

p−2

∂ν

j

∂Ω

j

d

(14)

(15)

## The Variational Principles of Bolle and Tanaka

2

2

1

(16)

θ

θ0

B,A

A

A

0

B,A

g∈SB,A

g(B)

0

n

n

n∈N

θ0n

n

E∗

θn

n

θ

θ

θ0

1

2

1

2

θ

1

θ

θ

2

θ

(17)

0

||u||→+∞, u∈B

θ∈[0,1]

θ

B,A

A

1

2

1

2

i

∂θ

i

i

i

1

2

1

2

1

2

i

1

2

1

i

1

2

i

2

1

2

0

1

0

2

0

0

1

2

1

2

0

0

1

2

0

i

1

i

2

i

1

2

i

1

2

i

i

1

0

2

0

1

2

Li0−τ )

(18)

1

0

2

0

2

1

B,A

2

A

1

1

B,A

B,A

A

1

2

1

2

B,A

A

1

2

θ

B,A

A

1

2

B,A

1

B,A

2

A

2

A

1

B,A

0

B,A

A

B,A

1

V

### = 0 and

1To see why note that for every t ∈ [0, 1] and x ∈ A we have H(t, x) /∈ D. So since D is closed there exists an interval Itcentered at t and a neighborhood Uxof x such that H(It, Ux) ∩ D = ∅. Since such sets cover [0, 1] × A we get H([0, 1], U ) ∩ D = ∅, where U = ∪xUx.

(19)

E\U

B,A

B,A

−1

−1

−1

−1

−1

0

0

0

B0,A0

0

0

0

B0,A0

0

0

θ

θ0

1

θ

2

θ

(20)

θ

1

2

1

∂θ

1

1

1

2

∂θ

2

2

2

i

2

A

1

B,A

2

A

1

B,A

1

1

B,A

2

2

A

1

2

1

2

1

2

1

### (R, [0, 1]) be

1To be more explicit let τ = 1 − θ. Then ∂τϕ1 = −∂θϕ1 = −f1(1 − τ, ϕ1) + δ. Similarly ∂τϕ2 =

−f2(1 − τ, ϕ2) − δ. Since ϕ1|τ =0> ϕ2|τ =0the comparison theorem implies that ϕ1> ϕ2 for all τ ∈ [0, 1].

(21)

θ

θ0

1

θ

2

θ

1

θ

1+

1

θ

θ0

E∗

θ0

θ0

2

2

θ+

2

2

θ

θ0

E∗

θ0

θ0

2

+

θ0

i

∂θ

i

i

i

i

θ

θ0

i

E

E

i±

i

(22)

i

i

i

−1

i

i

2

1

2

1

0

A

θ

2

2

0

A

θ

2

A

2

2

0

02

2

0

2

0

θ

2

θ0

2

2

2

(23)

0

θ

2

θ+

2

2

2

θ

2

θ0

2

t∈[0,1]

2

2

1

t∈[0,1]

1

2

θ0

2

θ

2

2

θ

2

2

0

θ

2

2

2

2

02

θ0

2

θ0

2

2

0

θ

2

θ+

2

2

2

2

2

02

0

B,A

θ

1

1

(24)

0

A

0

B,A

θ

2

2

1

θ

1

2

1

0

2

0

2

0

1

2

0

0

B0,A0

0

0

g∈S

B0,A0

g(B0)

1

D0

1

D0

1

1

2

A0

1

g∈SB0,A0

g(B0)

1

A0

1

1

g∈SB0,A0

g(B0)

1

1

g∈SB0,A0

g(B0)

1

1

1

B,A

(25)

1

B,A

1

B,A

g∈SB0,A0

g(B0)

1

1

B,A

g∈SB0,A0

g(B0)

1

1

B,A

1

B,A

1

B,A

1

B,A

1

1

B,A

1

B,A

2

B,A

k

k=0

k

k

k−1

k

0

k

k

k

0

k

g∈Γk

u∈g(Ek)

0

(26)

0

k

k

0

i

i

i

θ∈[0,1]

i

2

0

θ∈[0,1]

θ

1

k

2

k

1

k+1

k

k+1

k

1

k+1

2

k

2

k

1

k+1

2

k

1

k+1

k

g(Ek)

0

k

k+1+

k

+

k+1

k

k

k

k+1+

k

k

(27)

1

Bk,Ak

2

Ak

0

Bk,Ak

h∈SBk,Ak

h(B

k)

0

Bk,Ak

k+1

Bk,Ak

E+

k+1

k

k

k

k+1

k+1

k+1

0

h(Bk)

0

m(E+k+1)

0

¯ m(Ek+1+ )

0

¯ m(Ek+1)

0

k+1

Bk,Ak

k+1

i

1

Bk,Ak

1

k+1

2

k

2

Ak

0

k

k

g(E

k)

0

k

1

k

2

k

1

k+1

k

2

k

1

k+1

i

i

i

i

i

(28)

i

k+1

k

1

k+1

1

1

k+1

1

2

k

2

2

k

2

1

1

k+1

2

2

k

1

2

i

2

1

0

0

[0,1]

0

0

0

0

0

i

i

i

i

i

i

i

i

i

2

i

i

i

2

0

0

0

i

i

2

0

i

i

0

−Liθ

i

−Liθ

i

−Liθ

−Liθ

0

i

−Liθ

0

0

−Liθ0

−Lia

−Lia

−Liθ0

i

i

(29)

0

Li0−a)

Li0−a)

i

i

Li

Li

i

i

i

i

i

i

k

g∈Γk

u∈g(Ek)

(30)

k

k

k

k

k

k=1

k

k

k−1

k

0

2

0

0

(31)

k

k

k

0

k

E

k

m

k

k

m

k

Em

0

k

Em

k

k

k

k

k

Ek

0

k

E

k

k

k

k

k

k

k

g∈Γk

u∈g(Ek)

k

(32)

m

+

k

k

k

0

k

0

00

k

0

00

00

0

00

00

k

k

m

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