On degenerate $(q,p)$-Laplace equations corresponding to an inverse spectral problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914646900867072 |
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| author | Il'yasov, Y. Sh. Valeev, N. F. |
| author_facet | Il'yasov, Y. Sh. Valeev, N. F. |
| contents | Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded coefficients is introduced. In the main result, the existence, uniqueness, and stability of the nonnegative weak solution to the equations of the form
$$
-{\rm div}(ρ|\nabla u|^{q-2} \nabla u)-{\rm div}(|\nabla u|^{p-2}\nabla u)=λb |u|^{q-2}u,~~p>q
$$
are proven. Additionally, an explicit formula that expresses the solution of the equation through the inverse optimal solution of the spectral problem
$$-{\rm div}(ρ|\nabla ϕ|^{q-2}\nabla ϕ)=λb|ϕ|^{q-2}ϕ$$
is presented. The advantage of the method is that the inverse optimal problem has a visible geometry and a simple variational structure, which makes it easy to solve it and, as a consequence, find a solution to the associated nonlinear boundary value problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11171 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On degenerate $(q,p)$-Laplace equations corresponding to an inverse spectral problem Il'yasov, Y. Sh. Valeev, N. F. Analysis of PDEs Mathematical Physics Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded coefficients is introduced. In the main result, the existence, uniqueness, and stability of the nonnegative weak solution to the equations of the form $$ -{\rm div}(ρ|\nabla u|^{q-2} \nabla u)-{\rm div}(|\nabla u|^{p-2}\nabla u)=λb |u|^{q-2}u,~~p>q $$ are proven. Additionally, an explicit formula that expresses the solution of the equation through the inverse optimal solution of the spectral problem $$-{\rm div}(ρ|\nabla ϕ|^{q-2}\nabla ϕ)=λb|ϕ|^{q-2}ϕ$$ is presented. The advantage of the method is that the inverse optimal problem has a visible geometry and a simple variational structure, which makes it easy to solve it and, as a consequence, find a solution to the associated nonlinear boundary value problem. |
| title | On degenerate $(q,p)$-Laplace equations corresponding to an inverse spectral problem |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2401.11171 |