The Snapshot Problem for the Euler-Poisson-Darboux Equation
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arXiv
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| Format: | Preprint |
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2025
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| author | Gonzalez, Fulton Wang, Jue Christensen, Jens Kakehi, Tomoyuki |
| author_facet | Gonzalez, Fulton Wang, Jue Christensen, Jens Kakehi, Tomoyuki |
| contents | The generalized Euler-Poisson-Darboux (EPD) equation with complex parameter $α$ is given by $$ Δ_x u=\frac{\partial^2 u}{\partial t^2}+\frac{n-1+2α}{t}\,\frac{\partial u}{\partial t}, $$ where $u(x,t)\in \mathscr E(\mathbb R^n\times \mathbb R)$, with $u$ even in $t$. For $α=0$ and $α=1$ the solution $u(x,t)$ represents a mean value over spheres and balls, respectively, of radius $|t|$ in $\mathbb R^n$. In this paper we consider existence and uniqueness results for the following two-snapshot problem: for fixed positive real numbers $r$ and $s$ and smooth functions $f$ and $g$ on $\mathbb R^n$, what are the conditions under which there is a solution $u(x,t)$ to the generalized EPD equation such that $u(x,r)=f(x)$ and $u(x,s)=g(x)$? The answer leads to a discovery of Liouville-like numbers related to Bessel functions, and we also study the properties of such numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Snapshot Problem for the Euler-Poisson-Darboux Equation Gonzalez, Fulton Wang, Jue Christensen, Jens Kakehi, Tomoyuki Analysis of PDEs 35L05 (Primary), 58J45 (Secondary), 43A85 (Secondary) The generalized Euler-Poisson-Darboux (EPD) equation with complex parameter $α$ is given by $$ Δ_x u=\frac{\partial^2 u}{\partial t^2}+\frac{n-1+2α}{t}\,\frac{\partial u}{\partial t}, $$ where $u(x,t)\in \mathscr E(\mathbb R^n\times \mathbb R)$, with $u$ even in $t$. For $α=0$ and $α=1$ the solution $u(x,t)$ represents a mean value over spheres and balls, respectively, of radius $|t|$ in $\mathbb R^n$. In this paper we consider existence and uniqueness results for the following two-snapshot problem: for fixed positive real numbers $r$ and $s$ and smooth functions $f$ and $g$ on $\mathbb R^n$, what are the conditions under which there is a solution $u(x,t)$ to the generalized EPD equation such that $u(x,r)=f(x)$ and $u(x,s)=g(x)$? The answer leads to a discovery of Liouville-like numbers related to Bessel functions, and we also study the properties of such numbers. |
| title | The Snapshot Problem for the Euler-Poisson-Darboux Equation |
| topic | Analysis of PDEs 35L05 (Primary), 58J45 (Secondary), 43A85 (Secondary) |
| url | https://arxiv.org/abs/2507.13257 |