Diffusion with nonlocal boundary conditions

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• 作者：Wolfgang Arendt^a ; ^{wolfgang.arendt@uni-ulm.de" class="auth_mail" title="E-mail the corresponding author} ; Stefan Kunkel^b ; ^{stefan.kunkel@uni-ulm.de" class="auth_mail" title="E-mail the corresponding author} ; Markus Kunze^b ; ^{markus.kunze@uni-ulm.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47D07 ; 60J35 ; 35B35
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：270
• 期：7
• 页码：2483-2507
• 全文大小：510 K

文摘

We consider second order differential operators mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si1.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=33db52c035bc2f9ff3a64490382f17db" title="Click to view the MathML source">A_μmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>Ami>mrow><mrow><mi>μmi>mrow>msub>math> on a bounded, Dirichlet regular set mmlsi13" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si13.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=2f5dc5a5468ab6aa09453bd5db580943" title="Click to view the MathML source">Ω⊂R^dmathContainer hidden">mathCode"><math altimg="si13.gif" overflow="scroll"><mi mathvariant="normal">Ωmi><mo>⊂mo><msup><mrow><mi mathvariant="double-struck">Rmi>mrow><mrow><mi>dmi>mrow>msup>math>, subject to the nonlocal boundary conditions Here the function mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si4.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=04f91f3380ba8efcc2d0338b6a8f87de" title="Click to view the MathML source">μ:∂Ω→M⁺(Ω)mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><mi>μmi><mo>:mo><mo>∂mo><mi mathvariant="normal">Ωmi><mo stretchy="false">→mo><msup><mrow><mi mathvariant="script">Mmi>mrow><mrow><mo>+mo>mrow>msup><mo stretchy="false">(mo><mi mathvariant="normal">Ωmi><mo stretchy="false">)mo>math> is mmlsi258" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si258.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=6e63a011f3bda3149615857044fac4a5" title="Click to view the MathML source">σ(M(Ω),C_b(Ω))mathContainer hidden">mathCode"><math altimg="si258.gif" overflow="scroll"><mi>σmi><mo stretchy="false">(mo><mi mathvariant="script">Mmi><mo stretchy="false">(mo><mi mathvariant="normal">Ωmi><mo stretchy="false">)mo><mo>,mo><msub><mrow><mi>Cmi>mrow><mrow><mi>bmi>mrow>msub><mo stretchy="false">(mo><mi mathvariant="normal">Ωmi><mo stretchy="false">)mo><mo stretchy="false">)mo>math>-continuous with mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si6.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=58123725f5af0d306b2e1f4af7b5afbc" title="Click to view the MathML source">0≤μ(z,Ω)≤1mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mn>0mn><mo>≤mo><mi>μmi><mo stretchy="false">(mo><mi>zmi><mo>,mo><mi mathvariant="normal">Ωmi><mo stretchy="false">)mo><mo>≤mo><mn>1mn>math> for all mmlsi401" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si401.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=42b682328969e81130e79cbf7ba39a13" title="Click to view the MathML source">z∈∂ΩmathContainer hidden">mathCode"><math altimg="si401.gif" overflow="scroll"><mi>zmi><mo>∈mo><mo>∂mo><mi mathvariant="normal">Ωmi>math>. Under suitable assumptions on the coefficients in mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si1.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=33db52c035bc2f9ff3a64490382f17db" title="Click to view the MathML source">A_μmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>Ami>mrow><mrow><mi>μmi>mrow>msub>math>, we prove that mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si1.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=33db52c035bc2f9ff3a64490382f17db" title="Click to view the MathML source">A_μmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>Ami>mrow><mrow><mi>μmi>mrow>msub>math> generates a holomorphic positive contraction semigroup mmlsi47" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si47.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=e2e8c5c99f5e2d0e2844f3c49bbcfb9b" title="Click to view the MathML source">T_μmathContainer hidden">mathCode"><math altimg="si47.gif" overflow="scroll"><msub><mrow><mi>Tmi>mrow><mrow><mi>μmi>mrow>msub>math> on mmlsi42" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si42.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=3c78f99fd8ec7de73f12fce35f8c7854" title="Click to view the MathML source">L^∞(Ω)mathContainer hidden">mathCode"><math altimg="si42.gif" overflow="scroll"><msup><mrow><mi>Lmi>mrow><mrow><mo>∞mo>mrow>msup><mo stretchy="false">(mo><mi mathvariant="normal">Ωmi><mo stretchy="false">)mo>math>. The semigroup mmlsi47" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si47.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=e2e8c5c99f5e2d0e2844f3c49bbcfb9b" title="Click to view the MathML source">T_μmathContainer hidden">mathCode"><math altimg="si47.gif" overflow="scroll"><msub><mrow><mi>Tmi>mrow><mrow><mi>μmi>mrow>msub>math> is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in mmlsi10" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si10.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=692cb205d2cf506dafbb2e3480022cdb">mg class="imgLazyJSB inlineImage" height="20" width="36" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616000355-si10.gif">mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll"><mi>Cmi><mo stretchy="false">(mo><mover accent="true"><mrow><mi mathvariant="normal">Ωmi>mrow><mo>‾mo>mover><mo stretchy="false">)mo>math>. We also prove that mmlsi47" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si47.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=e2e8c5c99f5e2d0e2844f3c49bbcfb9b" title="Click to view the MathML source">T_μmathContainer hidden">mathCode"><math altimg="si47.gif" overflow="scroll"><msub><mrow><mi>Tmi>mrow><mrow><mi>μmi>mrow>msub>math> is immediately compact and study the asymptotic behavior of mmlsi11" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si11.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=7a98f428089632b4391a3986e85b20d9" title="Click to view the MathML source">T_μ(t)mathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll"><msub><mrow><mi>Tmi>mrow><mrow><mi>μmi>mrow>msub><mo stretchy="false">(mo><mi>tmi><mo stretchy="false">)mo>math> as mmlsi12" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000355&_mathId=si12.gif&_user=111111111&_pii=S0022123616000355&_rdoc=1&_issn=00221236&md5=b228485ea0e84ba6db3fc7d016fe66a8" title="Click to view the MathML source">t→∞mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll"><mi>tmi><mo stretchy="false">→mo><mo>∞mo>math>.