Local Lipschitz continuity of the inverse of the fractional -Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains

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• 作者：Mahamadi Warma ^{mahamadi.warma1@upr.edu" class="auth_mail" title="E-mail the corresponding author mjwarma@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35R11 ; 35J60 ; 35K55 ; 35B65 ; 47J30
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：135
• 期：Complete
• 页码：129-157
• 全文大小：771 K

文摘

Let p∈(1,∞)$p\in (1,\infty )$, 78" title="Click to view the MathML source">s∈(0,1)$s\in (0,1)$ and Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$ an arbitrary bounded open set. In the first part we consider the inverse ${\Phi }_{s,p}:={\left[{(-\Delta )}_{p,\Omega }^s\right]}^{-1}$ of the fractional 78e4753b12ba0c3" title="Click to view the MathML source">p$p$-Laplace operator ${(-\Delta )}_{p,\Omega }^s$ with the Dirichlet boundary condition. We show that in the singular case 78d008a008cb" title="Click to view the MathML source">p∈(1,2)$p\in (1,2)$, the operator Φ_s,p${\Phi }_{s,p}$ is locally Lipschitz continuous on e75a3faabfa2c83ca77508" title="Click to view the MathML source">L^∞(Ω)$L^{\infty }\left(\Omega \right)$ and that global Lipschitz continuity cannot be achieved. We use this result to show that in the case N>sp$N>sp$, if $\frac{2N}{N+2s}<p<2$, $0\le q\le \frac{Np}{N-sp}-2$ and e74f939b1dc7ad" title="Click to view the MathML source">α,β$\alpha ,\beta$ are small constants, then the nonlinear problem has at least one weak solution. In the second part of the paper, we prove that the operator $-{(-\Delta )}_{p,\Omega }^s$ generates a (nonlinear) submarkovian semigroup (S_s,p(t))_t≥0${\left(S_{s,p}\left(t\right)\right)}_{t\ge 0}$ on L²(Ω)$L^2\left(\Omega \right)$. If p∈[2,∞)$p\in [2,\infty )$ and sp<N$sp<N$, we obtain that 78" title="Click to view the MathML source">S_s,p(t)$S_{s,p}\left(t\right)$ satisfies the following e7d" title="Click to view the MathML source">(L^q−L^∞)$(L^q-L^{\infty })$-Hölder type estimate: there exists a constant C>0$C>0$ such that for every t>0$t>0$ and 78a9adedce3c0649cfe47e84" title="Click to view the MathML source">u,v∈L^q(Ω)$u,v\in L^q\left(\Omega \right)$ (q≥2$q\ge 2$), we have where β(s),δ(s)$\beta \left(s\right),\delta \left(s\right)$ and 78f4b0f4e9c9a63d3a62332e1dfe7f4d" title="Click to view the MathML source">γ(s)$\gamma \left(s\right)$ are explicit constants depending only on N,s,p,q$N,s,p,q$.