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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| Abstract | 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. |
|---|---|
| AbstractList | We prove sharp regularity results for a general class of functionals of the type w↦∫F(x,w,Dw)dx,featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral w↦∫b(x,w)(|Dw|p+a(x)|Dw|q)dx,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. 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. |
| ArticleNumber | 62 |
| Author | Mingione, Giuseppe Baroni, Paolo Colombo, Maria |
| Author_xml | – sequence: 1 givenname: Paolo surname: Baroni fullname: Baroni, Paolo organization: Dipartimento SMFI, Università di Parma – sequence: 2 givenname: Maria surname: Colombo fullname: Colombo, Maria organization: Institute for Theoretical Studies, ETH Zürich – sequence: 3 givenname: Giuseppe surname: Mingione fullname: Mingione, Giuseppe email: rosariomingione@gmail.com organization: Dipartimento SMFI, Università di Parma |
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| ContentType | Journal Article |
| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature 2018 Copyright Springer Science & Business Media 2018 |
| Copyright_xml | – notice: Springer-Verlag GmbH Germany, part of Springer Nature 2018 – notice: Copyright Springer Science & Business Media 2018 |
| DBID | AAYXX CITATION JQ2 |
| DOI | 10.1007/s00526-018-1332-z |
| DatabaseName | CrossRef ProQuest Computer Science Collection |
| DatabaseTitle | CrossRef ProQuest Computer Science Collection |
| DatabaseTitleList | ProQuest Computer Science Collection |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1432-0835 |
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| Snippet | 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... We prove sharp regularity results for a general class of functionals of the type w↦∫F(x,w,Dw)dx,featuring non-standard growth conditions and non-uniform... |
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| SubjectTerms | Analysis Calculus of Variations and Optimal Control; Optimization Control Ellipticity Functionals Mathematical and Computational Physics Mathematics Mathematics and Statistics Regularity Systems Theory Theoretical |
| Title | Regularity for general functionals with double phase |
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