Loading…
A nonlinear elliptic boundary value problem relevant in general relativity and in the theory of electrical heating of conductors
The elliptic boundary value problem governing the steady electrical heating of a conductor of heat and electricity, the so-called thermistor problem, ∇ · ( σ ( u ) ∇ ϕ ) = 0 in Ω ϕ = ϕ b on Γ ∇ · ( κ ( u ) ∇ u ) = - σ ( u ) | ∇ ϕ | 2 in Ω u = 0 on Γ , where σ ( u ) is the temperature dependent elect...
Saved in:
| Published in: | Bollettino della Unione matematica italiana (2008) 2018-06, Vol.11 (2), p.191-204 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | The elliptic boundary value problem governing the steady electrical heating of a conductor of heat and electricity, the so-called thermistor problem,
∇
·
(
σ
(
u
)
∇
ϕ
)
=
0
in
Ω
ϕ
=
ϕ
b
on
Γ
∇
·
(
κ
(
u
)
∇
u
)
=
-
σ
(
u
)
|
∇
ϕ
|
2
in
Ω
u
=
0
on
Γ
,
where
σ
(
u
)
is the temperature dependent electric conductivity and
κ
(
u
)
the thermal conductivity, admits a reinterpretation in the framework of general relativity if we choose
σ
(
u
)
=
e
u
,
κ
(
u
)
=
1
and, in addition,
Ω
is a domain of
R
3
axially symmetric whereas the function
ϕ
b
, in a cylindrical coordinate system
ρ
,
z
,
φ
, is independent of
φ
. The same analytical methods relevant in the thermistor problem can be used in this new context. |
|---|---|
| ISSN: | 1972-6724 2198-2759 |
| DOI: | 10.1007/s40574-017-0121-5 |