The obstacle problem for a higher order fractional Laplacian
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911761359175680 |
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| author | Danielli, Donatella Ali, Alaa Haj Petrosyan, Arshak |
| author_facet | Danielli, Donatella Ali, Alaa Haj Petrosyan, Arshak |
| contents | In this paper, we consider the obstacle problem for the fractional Laplace operator $(-Δ)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper half-space $\mathbb{R}_+^{n+1}$ to obtain a thin obstacle problem for the weighted biLaplace operator $Δ^2_b U$, where $Δ_b U=y^{-b}\nabla \cdot (y^b \nabla U)$. Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and $C_{loc}^{1,1}(\R^n) \cap H^{1+s}(\R^n)$-regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in \cite{Sc1} and \cite{Sc2} from the case $b=0$ to the general case $-1<b<1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_07201 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The obstacle problem for a higher order fractional Laplacian Danielli, Donatella Ali, Alaa Haj Petrosyan, Arshak Analysis of PDEs 35B65, 35R35, 35J35, 35R11 In this paper, we consider the obstacle problem for the fractional Laplace operator $(-Δ)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper half-space $\mathbb{R}_+^{n+1}$ to obtain a thin obstacle problem for the weighted biLaplace operator $Δ^2_b U$, where $Δ_b U=y^{-b}\nabla \cdot (y^b \nabla U)$. Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and $C_{loc}^{1,1}(\R^n) \cap H^{1+s}(\R^n)$-regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in \cite{Sc1} and \cite{Sc2} from the case $b=0$ to the general case $-1<b<1$. |
| title | The obstacle problem for a higher order fractional Laplacian |
| topic | Analysis of PDEs 35B65, 35R35, 35J35, 35R11 |
| url | https://arxiv.org/abs/2305.07201 |