T. Wunderli
求助PDF
{"title":"一类泛函在l1上的下半连续性","authors":"T. Wunderli","doi":"10.1155/2021/6709303","DOIUrl":null,"url":null,"abstract":"<jats:p>We prove lower semicontinuity in <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msup>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> for a class of functionals <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mi mathvariant=\"script\">G</mi>\n <mo>:</mo>\n <mi>B</mi>\n <mi>V</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </mfenced>\n <mo>⟶</mo>\n <mi>ℝ</mi>\n </math>\n </jats:inline-formula> of the form <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi mathvariant=\"script\">G</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>u</mi>\n </mrow>\n </mfenced>\n <mo>=</mo>\n <msub>\n <mrow>\n <mo>∫</mo>\n </mrow>\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </msub>\n <mi>g</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n <mrow>\n <mo>,</mo>\n </mrow>\n <mrow>\n <mo>∇</mo>\n </mrow>\n <mi>u</mi>\n </mrow>\n </mfenced>\n <mi>d</mi>\n <mi>x</mi>\n <mo>+</mo>\n <msub>\n <mrow>\n <mo>∫</mo>\n </mrow>\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </msub>\n <mi>ψ</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n </mrow>\n </mfenced>\n <mi>d</mi>\n <mfenced open=\"|\" close=\"|\">\n <mrow>\n <msup>\n <mrow>\n <mi>D</mi>\n </mrow>\n <mrow>\n <mi>s</mi>\n </mrow>\n </msup>\n <mi>u</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> where <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <mi>g</mi>\n <mo>:</mo>\n <mi>Ω</mi>\n <mo>×</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mi>N</mi>\n </mrow>\n </msup>\n <mo>⟶</mo>\n <mi>ℝ</mi>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <mi>Ω</mi>\n <mo>⊂</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mi>N</mi>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> is open and bounded, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mi>g</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mo>·</mo>\n <mo>,</mo>\n <mi>p</mi>\n </mrow>\n </mfenced>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msup>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> for each <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <mi>p</mi>\n <mo>,</mo>\n </math>\n </jats:inline-formula> satisfies the linear growth condition <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <munder>\n <mrow>\n <mi mathvariant=\"normal\">lim</mi>\n </mrow>\n <mrow>\n <mfenced open=\"|\" close=\"|\">\n <mrow>\n <mi>p</mi>\n </mrow>\n </mfenced>\n <mrow>\n <mo>⟶</mo>\n </mrow>\n <mrow>\n <mo>∞</mo>\n </mrow>\n </mrow>\n </munder>\n <mi>g</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mi>p</mi>\n </mrow>\n </mfenced>\n <mo>/</mo>\n <mfenced open=\"|\" close=\"|\">\n <mrow>\n <mi>p</mi>\n </mrow>\n </mfenced>\n <mo>=</mo>\n <mi>ψ</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n </mrow>\n </mfenced>\n <mo>∈</mo>\n <mi>C</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </mfenced>\n <mo>∩</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mo>∞</mo>\n </mrow>\n </msup>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>Ω</mi>\n </mrow>\n </mfenced>\n <mo>,</mo>\n </math>\n </jats:inline-formula> and is convex in <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\">\n <mi>p</mi>\n </math>\n </jats:inline-formula> depending only on <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\">\n <mfenced open=\"|\" close=\"|\">\n <mrow>\n <mi>p</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> for a.e. <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M13\">\n <mi>x</mi>\n <mo>.</mo>\n </math>\n </jats:inline-formula> Here, we recall for <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M14\">\n <mi>u</mi>\n <mo>∈</mo>\n <mi>B</mi>\n <mi>V</mi>\n <mfenced open=\"(\" close=\")\">\n ","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Lower Semicontinuity in \\n \\n \\n L\\n \\n \\n 1\\n \\n \\n of a Class of Functionals Defined on \\n \",\"authors\":\"T. Wunderli\",\"doi\":\"10.1155/2021/6709303\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>We prove lower semicontinuity in <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M3\\\">\\n <msup>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> for a class of functionals <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M4\\\">\\n <mi mathvariant=\\\"script\\\">G</mi>\\n <mo>:</mo>\\n <mi>B</mi>\\n <mi>V</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </mfenced>\\n <mo>⟶</mo>\\n <mi>ℝ</mi>\\n </math>\\n </jats:inline-formula> of the form <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M5\\\">\\n <mi mathvariant=\\\"script\\\">G</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>u</mi>\\n </mrow>\\n </mfenced>\\n <mo>=</mo>\\n <msub>\\n <mrow>\\n <mo>∫</mo>\\n </mrow>\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </msub>\\n <mi>g</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>x</mi>\\n <mrow>\\n <mo>,</mo>\\n </mrow>\\n <mrow>\\n <mo>∇</mo>\\n </mrow>\\n <mi>u</mi>\\n </mrow>\\n </mfenced>\\n <mi>d</mi>\\n <mi>x</mi>\\n <mo>+</mo>\\n <msub>\\n <mrow>\\n <mo>∫</mo>\\n </mrow>\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </msub>\\n <mi>ψ</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>x</mi>\\n </mrow>\\n </mfenced>\\n <mi>d</mi>\\n <mfenced open=\\\"|\\\" close=\\\"|\\\">\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n <mrow>\\n <mi>s</mi>\\n </mrow>\\n </msup>\\n <mi>u</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> where <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M6\\\">\\n <mi>g</mi>\\n <mo>:</mo>\\n <mi>Ω</mi>\\n <mo>×</mo>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mi>N</mi>\\n </mrow>\\n </msup>\\n <mo>⟶</mo>\\n <mi>ℝ</mi>\\n </math>\\n </jats:inline-formula>, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M7\\\">\\n <mi>Ω</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mi>N</mi>\\n </mrow>\\n </msup>\\n </math>\\n </jats:inline-formula> is open and bounded, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M8\\\">\\n <mi>g</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mo>·</mo>\\n <mo>,</mo>\\n <mi>p</mi>\\n </mrow>\\n </mfenced>\\n <mo>∈</mo>\\n <msup>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> for each <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M9\\\">\\n <mi>p</mi>\\n <mo>,</mo>\\n </math>\\n </jats:inline-formula> satisfies the linear growth condition <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M10\\\">\\n <munder>\\n <mrow>\\n <mi mathvariant=\\\"normal\\\">lim</mi>\\n </mrow>\\n <mrow>\\n <mfenced open=\\\"|\\\" close=\\\"|\\\">\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </mfenced>\\n <mrow>\\n <mo>⟶</mo>\\n </mrow>\\n <mrow>\\n <mo>∞</mo>\\n </mrow>\\n </mrow>\\n </munder>\\n <mi>g</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>p</mi>\\n </mrow>\\n </mfenced>\\n <mo>/</mo>\\n <mfenced open=\\\"|\\\" close=\\\"|\\\">\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </mfenced>\\n <mo>=</mo>\\n <mi>ψ</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>x</mi>\\n </mrow>\\n </mfenced>\\n <mo>∈</mo>\\n <mi>C</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </mfenced>\\n <mo>∩</mo>\\n <msup>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mo>∞</mo>\\n </mrow>\\n </msup>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>Ω</mi>\\n </mrow>\\n </mfenced>\\n <mo>,</mo>\\n </math>\\n </jats:inline-formula> and is convex in <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M11\\\">\\n <mi>p</mi>\\n </math>\\n </jats:inline-formula> depending only on <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M12\\\">\\n <mfenced open=\\\"|\\\" close=\\\"|\\\">\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> for a.e. <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M13\\\">\\n <mi>x</mi>\\n <mo>.</mo>\\n </math>\\n </jats:inline-formula> Here, we recall for <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M14\\\">\\n <mi>u</mi>\\n <mo>∈</mo>\\n <mi>B</mi>\\n <mi>V</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n \",\"PeriodicalId\":7061,\"journal\":{\"name\":\"Abstract and Applied Analysis\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-11-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Abstract and Applied Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1155/2021/6709303\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abstract and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2021/6709303","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
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Lower Semicontinuity in
L
1
of a Class of Functionals Defined on
We prove lower semicontinuity in
L
1
Ω
for a class of functionals
G
:
B
V
Ω
⟶
ℝ
of the form
G
u
=
∫
Ω
g
x
,
∇
u
d
x
+
∫
Ω
ψ
x
d
D
s
u
where
g
:
Ω
×
ℝ
N
⟶
ℝ
,
Ω
⊂
ℝ
N
is open and bounded,
g
·
,
p
∈
L
1
Ω
for each
p
,
satisfies the linear growth condition
lim
p
⟶
∞
g
x
,
p
/
p
=
ψ
x
∈
C
Ω
∩
L
∞
Ω
,
and is convex in
p
depending only on
p
for a.e.
x
.
Here, we recall for
u
∈
B
V