Maximum principles, extension problem and inversion for nonlocal one-sided equations

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• 作者：Ana Bernardis^a ; ^{bernard@santafe-conicet.gov.ar" class="auth_mail" title="E-mail the corresponding author} ; Francisco J. Martí ; n-Reyes^b ; ^{martin_reyes@uma.es" class="auth_mail" title="E-mail the corresponding author} ; Pablo Raú ; l Stinga^c ; ^{stinga@iastate.edu" class="auth_mail" title="E-mail the corresponding author} ; José ; L. Torrea^d ; ^{joseluis.torrea@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35R11 ; 34A08 ; secondary ; 26A33 ; 35A08 ; 35B50
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：6333-6362
• 全文大小：413 K

文摘

We study one-sided nonlocal equations of the form on the real line. Notice that to compute this nonlocal operator of order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si2.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=a33b11162d8a9f7234d6b161739b95ad" title="Click to view the MathML source">0<α<1class="mathContainer hidden">class="mathCode">$0<\alpha <1$ at a point class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si3.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=30a056582f96dfd44ab7c779d2ddfdf3" title="Click to view the MathML source">x₀class="mathContainer hidden">class="mathCode">$x_0$ we need to know the values of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si4.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=31fe1a1816e0a120816c8be0ddadf952" title="Click to view the MathML source">u(x)class="mathContainer hidden">class="mathCode">$u(x)$ to the right of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si3.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=30a056582f96dfd44ab7c779d2ddfdf3" title="Click to view the MathML source">x₀class="mathContainer hidden">class="mathCode">$x_0$, that is, for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si38.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=51ce4e2ed899afab61c4b5262827e01c" title="Click to view the MathML source">x≥x₀class="mathContainer hidden">class="mathCode">$x\ge x_0$. We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli–Silvestre and Stinga–Torrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.