On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data
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| Format: | Preprint |
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2025
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| author | Ghosh, Sekhar Lakshmi, R. Zhang, Chao |
| author_facet | Ghosh, Sekhar Lakshmi, R. Zhang, Chao |
| contents | This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*}
L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } Ω, \end{align*} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{align*}
L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y)
&\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_16461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data Ghosh, Sekhar Lakshmi, R. Zhang, Chao Analysis of PDEs 35D30, 35D40, 35R05, 35R11, 35B65 This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } Ω, \end{align*} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{align*} L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y) &\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes. |
| title | On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data |
| topic | Analysis of PDEs 35D30, 35D40, 35R05, 35R11, 35B65 |
| url | https://arxiv.org/abs/2505.16461 |