On Runge type theorems for solutions to strongly uniformly parabolic operators
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916435681345536 |
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| author | Vilkov, P. Yu. Shlapunov, A. A. |
| author_facet | Vilkov, P. Yu. Shlapunov, A. A. |
| contents | Let $G_1, G_2 $ be domains in ${\mathbb R}^{n+1}$, $n \geq 2$, such that $G_1 \subset G_2$ and the domain $G_1$ have rather regular boundary. We investigate the problem of approximation of solutions to strongly uniformly $2m$-parabolic system $\mathcal L$ in the domain $G_1$ by solutions to the same system in the domain $G_2$. First, we prove that the space $S _{\mathcal L}(G_2)$ of solutions to the system $\mathcal L$ in the domain $G_2$ is dense in the space $S _{\mathcal L}(G_1)$, endowed with the standard Fréchet topology of the uniform convergence on compact subsets in $G_1$, if and only if the complements $G_2 (t) \setminus G_1 (t)$ have no non-empty compact components in $G_2 (t)$ for each $t\in \mathbb R$, where $G_j (t) = \{x \in {\mathbb R}^n: (x,t) \in G_j\}$. Next, under additional assumptions on the regularity of the bounded domains $G_1$ and $G_1(t)$, we prove that solutions from the Lebesgue class $L^2(G_1)\cap S _{\mathcal L}(G_1)$ can be approximated by solutions from $S _{\mathcal L}(G_2)$ if and only if the same assumption on the complements $G_2 (t) \setminus G_1 (t)$, $t\in \mathbb R$, is fulfilled. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18060 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Runge type theorems for solutions to strongly uniformly parabolic operators Vilkov, P. Yu. Shlapunov, A. A. Analysis of PDEs Primary 35K07, Secondary 35K25 Let $G_1, G_2 $ be domains in ${\mathbb R}^{n+1}$, $n \geq 2$, such that $G_1 \subset G_2$ and the domain $G_1$ have rather regular boundary. We investigate the problem of approximation of solutions to strongly uniformly $2m$-parabolic system $\mathcal L$ in the domain $G_1$ by solutions to the same system in the domain $G_2$. First, we prove that the space $S _{\mathcal L}(G_2)$ of solutions to the system $\mathcal L$ in the domain $G_2$ is dense in the space $S _{\mathcal L}(G_1)$, endowed with the standard Fréchet topology of the uniform convergence on compact subsets in $G_1$, if and only if the complements $G_2 (t) \setminus G_1 (t)$ have no non-empty compact components in $G_2 (t)$ for each $t\in \mathbb R$, where $G_j (t) = \{x \in {\mathbb R}^n: (x,t) \in G_j\}$. Next, under additional assumptions on the regularity of the bounded domains $G_1$ and $G_1(t)$, we prove that solutions from the Lebesgue class $L^2(G_1)\cap S _{\mathcal L}(G_1)$ can be approximated by solutions from $S _{\mathcal L}(G_2)$ if and only if the same assumption on the complements $G_2 (t) \setminus G_1 (t)$, $t\in \mathbb R$, is fulfilled. |
| title | On Runge type theorems for solutions to strongly uniformly parabolic operators |
| topic | Analysis of PDEs Primary 35K07, Secondary 35K25 |
| url | https://arxiv.org/abs/2310.18060 |