New applications of Hadamard-in-the-mean inequalities to incompressible variational problems New applications of Hadamard-in-the-mean inequalities
Let D ( u ) be the Dirichlet energy of a map u belonging to the Sobolev space H u 0 1 ( Ω ; R 2 ) and let A be a subclass of H u 0 1 ( Ω ; R 2 ) whose members are subject to the constraint det ∇ u = g a.e. for a given g , together with some boundary data u 0 . We develop a technique that, when appli...
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| Published in | Calculus of variations and partial differential equations Vol. 64; no. 8 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
17.09.2025
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0944-2669 1432-0835 |
| DOI | 10.1007/s00526-025-03119-x |
Cover
| Summary: | Let
D
(
u
)
be the Dirichlet energy of a map
u
belonging to the Sobolev space
H
u
0
1
(
Ω
;
R
2
)
and let
A
be a subclass of
H
u
0
1
(
Ω
;
R
2
)
whose members are subject to the constraint
det
∇
u
=
g
a.e. for a given
g
, together with some boundary data
u
0
. We develop a technique that, when applicable, enables us to characterize the global minimizer of
D
(
u
)
in
A
as the unique global minimizer of the associated functional
F
(
u
)
:
=
D
(
u
)
+
∫
Ω
f
(
x
)
det
∇
u
(
x
)
d
x
in the free class
H
u
0
1
(
Ω
;
R
2
)
. A key ingredient is the mean coercivity of
F
on
H
0
1
(
Ω
;
R
2
)
, which condition holds provided the ‘pressure’
f
∈
L
∞
(
Ω
)
is ‘tuned’ according to the procedure set out in [
1
]. The explicit examples to which our technique applies can be interpreted as solving the sort of constrained minimization problem that typically arises in incompressible nonlinear elasticity theory. |
|---|---|
| ISSN: | 0944-2669 1432-0835 |
| DOI: | 10.1007/s00526-025-03119-x |