Reconstruction of coefficients in the double phase problem
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arXiv
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| Format: | Preprint |
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2025
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| author | Cârstea, Cătălin I. Zimmermann, Philipp |
| author_facet | Cârstea, Cătălin I. Zimmermann, Philipp |
| contents | The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $Ω$, $u=f$ on $\partialΩ$, from the Dirichlet to Neumann (DN) map $Λ_a$. We show that this can be achieved, when the coefficient $a$ has Hölder continuous first order derivatives and the exponents satisfy $1<p\neq q<\infty$. Our reconstruction method relies on a careful analysis of the asymptotic behavior of the solution $u$ to the double phase problem with small or large Dirichlet datum $f$ (depending on the ordering of $p$ and $q$) as well as the related DN map $Λ_a$. As is common for inverse boundary value problems, we need a sufficiently rich family of special solutions to a related partial differential equation, which is independent of the coefficient one aims to reconstruct (in our case to the $p$-Laplace equation). We construct such families of solutions by a suitable linearization technique. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reconstruction of coefficients in the double phase problem Cârstea, Cătălin I. Zimmermann, Philipp Analysis of PDEs Primary 35R30, secondary 35J62, 35J70 The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $Ω$, $u=f$ on $\partialΩ$, from the Dirichlet to Neumann (DN) map $Λ_a$. We show that this can be achieved, when the coefficient $a$ has Hölder continuous first order derivatives and the exponents satisfy $1<p\neq q<\infty$. Our reconstruction method relies on a careful analysis of the asymptotic behavior of the solution $u$ to the double phase problem with small or large Dirichlet datum $f$ (depending on the ordering of $p$ and $q$) as well as the related DN map $Λ_a$. As is common for inverse boundary value problems, we need a sufficiently rich family of special solutions to a related partial differential equation, which is independent of the coefficient one aims to reconstruct (in our case to the $p$-Laplace equation). We construct such families of solutions by a suitable linearization technique. |
| title | Reconstruction of coefficients in the double phase problem |
| topic | Analysis of PDEs Primary 35R30, secondary 35J62, 35J70 |
| url | https://arxiv.org/abs/2504.01691 |