Regularity for general functionals with double phase
We prove sharp regularity results for a general class of functionals of the type w ↦ ∫ F ( x , w , D w ) d x , featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral w ↦ ∫ b ( x , w ) ( | D w | p + a ( x ) | D w | q ) d x...
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| Published in: | Calculus of variations and partial differential equations Vol. 57; no. 2; pp. 1 - 48 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.04.2018
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0944-2669, 1432-0835 |
| Online Access: | Get full text |
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| Summary: | We prove sharp regularity results for a general class of functionals of the type
w
↦
∫
F
(
x
,
w
,
D
w
)
d
x
,
featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral
w
↦
∫
b
(
x
,
w
)
(
|
D
w
|
p
+
a
(
x
)
|
D
w
|
q
)
d
x
,
1
<
p
<
q
,
a
(
x
)
≥
0
,
with
0
<
ν
≤
b
(
·
)
≤
L
. This changes its ellipticity rate according to the geometry of the level set
{
a
(
x
)
=
0
}
of the modulating coefficient
a
(
·
)
. We also present new methods and proofs that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0944-2669 1432-0835 |
| DOI: | 10.1007/s00526-018-1332-z |