Higher Order Boundary Harnack Principle via Degenerate Equations
As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type - div ρ a A ∇ w = ρ a f + div ρ a F in Ω for exponents a > - 1 , where the weight ρ vanishes with non zero gradient on a regular hypersurface Γ , which can be either a part o...
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| Veröffentlicht in: | Archive for rational mechanics and analysis 2024-04, Vol.248 (2), p.29, Article 29 |
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| creator | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| description | As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type
-
div
ρ
a
A
∇
w
=
ρ
a
f
+
div
ρ
a
F
in
Ω
for exponents
a
>
-
1
, where the weight
ρ
vanishes with non zero gradient on a regular hypersurface
Γ
, which can be either a part of the boundary of
Ω
or mostly contained in its interior. As an application, we extend such estimates to the ratio
v
/
u
of two solutions to a second order elliptic equation in divergence form when the zero set of
v
includes the zero set of
u
which is not singular in the domain (in this case
ρ
=
u
,
a
=
2
and
w
=
v
/
u
). We prove first the
C
k
,
α
-regularity of the ratio from one side of the regular part of the nodal set of
u
in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension
n
=
2
, we provide local gradient estimates for the ratio, which hold also across the singular set. |
| doi_str_mv | 10.1007/s00205-024-01973-1 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3015064000</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3015064000</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-99bfd70cc631d68e806eb5cf7ca7e88eb9406060b341ea55075101c86d24a83c3</originalsourceid><addsrcrecordid>eNp9kNFLwzAQxoMoOKf_gE8Fn6OXpEnaN3VOKwzmgz6HNL3OztluSSvsvzezgm9ycMfB9x33_Qi5ZHDNAPRNAOAgKfCUAsu1oOyITFgqOAWlxTGZAICgueT6lJyFsD6sXKgJuS2a1Tv6ZOmr2O-7oa2s3yeF9a11H8mLb1rXbDeYfDU2ecAVtuhtj8l8N9i-6dpwTk5quwl48Tun5O1x_jor6GL59Dy7W1AnWN7TPC_rSoNzSrBKZZiBwlK6WjurMcuwzFNQsUqRMrRSgpYMmMtUxVObCSem5Gq8u_XdbsDQm3U3xCc3wQhgElR6yDQlfFQ534XgsTZb33zGRIaBOZAyIykTSZkfUoZFkxhNIYrbFfq_0_-4vgEkhGos</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3015064000</pqid></control><display><type>article</type><title>Higher Order Boundary Harnack Principle via Degenerate Equations</title><source>Springer Nature - Complete Springer Journals</source><creator>Terracini, Susanna ; Tortone, Giorgio ; Vita, Stefano</creator><creatorcontrib>Terracini, Susanna ; Tortone, Giorgio ; Vita, Stefano</creatorcontrib><description>As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type
-
div
ρ
a
A
∇
w
=
ρ
a
f
+
div
ρ
a
F
in
Ω
for exponents
a
>
-
1
, where the weight
ρ
vanishes with non zero gradient on a regular hypersurface
Γ
, which can be either a part of the boundary of
Ω
or mostly contained in its interior. As an application, we extend such estimates to the ratio
v
/
u
of two solutions to a second order elliptic equation in divergence form when the zero set of
v
includes the zero set of
u
which is not singular in the domain (in this case
ρ
=
u
,
a
=
2
and
w
=
v
/
u
). We prove first the
C
k
,
α
-regularity of the ratio from one side of the regular part of the nodal set of
u
in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension
n
=
2
, we provide local gradient estimates for the ratio, which hold also across the singular set.</description><identifier>ISSN: 0003-9527</identifier><identifier>EISSN: 1432-0673</identifier><identifier>DOI: 10.1007/s00205-024-01973-1</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Classical Mechanics ; Complex Systems ; Conformal mapping ; Digital Object Identifier ; Divergence ; Elliptic functions ; Estimates ; Fluid- and Aerodynamics ; Hyperspaces ; Mathematical analysis ; Mathematical and Computational Physics ; Physics ; Physics and Astronomy ; Principles ; Theoretical</subject><ispartof>Archive for rational mechanics and analysis, 2024-04, Vol.248 (2), p.29, Article 29</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-99bfd70cc631d68e806eb5cf7ca7e88eb9406060b341ea55075101c86d24a83c3</citedby><cites>FETCH-LOGICAL-c319t-99bfd70cc631d68e806eb5cf7ca7e88eb9406060b341ea55075101c86d24a83c3</cites><orcidid>0000-0002-6846-7841</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00205-024-01973-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00205-024-01973-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51298</link.rule.ids></links><search><creatorcontrib>Terracini, Susanna</creatorcontrib><creatorcontrib>Tortone, Giorgio</creatorcontrib><creatorcontrib>Vita, Stefano</creatorcontrib><title>Higher Order Boundary Harnack Principle via Degenerate Equations</title><title>Archive for rational mechanics and analysis</title><addtitle>Arch Rational Mech Anal</addtitle><description>As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type
-
div
ρ
a
A
∇
w
=
ρ
a
f
+
div
ρ
a
F
in
Ω
for exponents
a
>
-
1
, where the weight
ρ
vanishes with non zero gradient on a regular hypersurface
Γ
, which can be either a part of the boundary of
Ω
or mostly contained in its interior. As an application, we extend such estimates to the ratio
v
/
u
of two solutions to a second order elliptic equation in divergence form when the zero set of
v
includes the zero set of
u
which is not singular in the domain (in this case
ρ
=
u
,
a
=
2
and
w
=
v
/
u
). We prove first the
C
k
,
α
-regularity of the ratio from one side of the regular part of the nodal set of
u
in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension
n
=
2
, we provide local gradient estimates for the ratio, which hold also across the singular set.</description><subject>Classical Mechanics</subject><subject>Complex Systems</subject><subject>Conformal mapping</subject><subject>Digital Object Identifier</subject><subject>Divergence</subject><subject>Elliptic functions</subject><subject>Estimates</subject><subject>Fluid- and Aerodynamics</subject><subject>Hyperspaces</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Principles</subject><subject>Theoretical</subject><issn>0003-9527</issn><issn>1432-0673</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kNFLwzAQxoMoOKf_gE8Fn6OXpEnaN3VOKwzmgz6HNL3OztluSSvsvzezgm9ycMfB9x33_Qi5ZHDNAPRNAOAgKfCUAsu1oOyITFgqOAWlxTGZAICgueT6lJyFsD6sXKgJuS2a1Tv6ZOmr2O-7oa2s3yeF9a11H8mLb1rXbDeYfDU2ecAVtuhtj8l8N9i-6dpwTk5quwl48Tun5O1x_jor6GL59Dy7W1AnWN7TPC_rSoNzSrBKZZiBwlK6WjurMcuwzFNQsUqRMrRSgpYMmMtUxVObCSem5Gq8u_XdbsDQm3U3xCc3wQhgElR6yDQlfFQ534XgsTZb33zGRIaBOZAyIykTSZkfUoZFkxhNIYrbFfq_0_-4vgEkhGos</recordid><startdate>20240401</startdate><enddate>20240401</enddate><creator>Terracini, Susanna</creator><creator>Tortone, Giorgio</creator><creator>Vita, Stefano</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0002-6846-7841</orcidid></search><sort><creationdate>20240401</creationdate><title>Higher Order Boundary Harnack Principle via Degenerate Equations</title><author>Terracini, Susanna ; Tortone, Giorgio ; Vita, Stefano</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-99bfd70cc631d68e806eb5cf7ca7e88eb9406060b341ea55075101c86d24a83c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Classical Mechanics</topic><topic>Complex Systems</topic><topic>Conformal mapping</topic><topic>Digital Object Identifier</topic><topic>Divergence</topic><topic>Elliptic functions</topic><topic>Estimates</topic><topic>Fluid- and Aerodynamics</topic><topic>Hyperspaces</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Principles</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Terracini, Susanna</creatorcontrib><creatorcontrib>Tortone, Giorgio</creatorcontrib><creatorcontrib>Vita, Stefano</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Archive for rational mechanics and analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Terracini, Susanna</au><au>Tortone, Giorgio</au><au>Vita, Stefano</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Higher Order Boundary Harnack Principle via Degenerate Equations</atitle><jtitle>Archive for rational mechanics and analysis</jtitle><stitle>Arch Rational Mech Anal</stitle><date>2024-04-01</date><risdate>2024</risdate><volume>248</volume><issue>2</issue><spage>29</spage><pages>29-</pages><artnum>29</artnum><issn>0003-9527</issn><eissn>1432-0673</eissn><abstract>As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type
-
div
ρ
a
A
∇
w
=
ρ
a
f
+
div
ρ
a
F
in
Ω
for exponents
a
>
-
1
, where the weight
ρ
vanishes with non zero gradient on a regular hypersurface
Γ
, which can be either a part of the boundary of
Ω
or mostly contained in its interior. As an application, we extend such estimates to the ratio
v
/
u
of two solutions to a second order elliptic equation in divergence form when the zero set of
v
includes the zero set of
u
which is not singular in the domain (in this case
ρ
=
u
,
a
=
2
and
w
=
v
/
u
). We prove first the
C
k
,
α
-regularity of the ratio from one side of the regular part of the nodal set of
u
in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension
n
=
2
, we provide local gradient estimates for the ratio, which hold also across the singular set.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00205-024-01973-1</doi><orcidid>https://orcid.org/0000-0002-6846-7841</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0003-9527 |
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| issn | 0003-9527 1432-0673 |
| language | eng |
| recordid | cdi_proquest_journals_3015064000 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Classical Mechanics Complex Systems Conformal mapping Digital Object Identifier Divergence Elliptic functions Estimates Fluid- and Aerodynamics Hyperspaces Mathematical analysis Mathematical and Computational Physics Physics Physics and Astronomy Principles Theoretical |
| title | Higher Order Boundary Harnack Principle via Degenerate Equations |
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