Global Leray-Schauder continuation for Fredholm operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908700344582144 |
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| author | López-Gómez, Julián Sampedro, Juan Carlos |
| author_facet | López-Gómez, Julián Sampedro, Juan Carlos |
| contents | This paper ascertains the global behavior of the forward and backward branches of solutions provided by the Leray-Schauder continuation theorem for orientable $\mathcal{C}^1$ Fredholm maps, as developed by the authors in [54]. Under properness on bounded sets and a nonzero local index at the given base solution, each branch satisfies the following alternative: either it is unbounded, or it reaches the boundary of the domain, or it accumulates at a different solution on the base parameter level. When the component is bounded and stays in the interior, there is a degree balance on the base slice entailing a vanishing sum of local indices and, in particular, the existence of an even number of non-degenerate contact points. For real-analytic maps we construct locally injective parameterizations that exhibit blow-up, approach to the boundary, or return to the base level. An application to a quasilinear boundary value problem driven by the mean-curvature and Minkowski operators illustrates the global results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Leray-Schauder continuation for Fredholm operators López-Gómez, Julián Sampedro, Juan Carlos Analysis of PDEs Functional Analysis 46T20, 47H11, 47A53, 46G12, 34K18 This paper ascertains the global behavior of the forward and backward branches of solutions provided by the Leray-Schauder continuation theorem for orientable $\mathcal{C}^1$ Fredholm maps, as developed by the authors in [54]. Under properness on bounded sets and a nonzero local index at the given base solution, each branch satisfies the following alternative: either it is unbounded, or it reaches the boundary of the domain, or it accumulates at a different solution on the base parameter level. When the component is bounded and stays in the interior, there is a degree balance on the base slice entailing a vanishing sum of local indices and, in particular, the existence of an even number of non-degenerate contact points. For real-analytic maps we construct locally injective parameterizations that exhibit blow-up, approach to the boundary, or return to the base level. An application to a quasilinear boundary value problem driven by the mean-curvature and Minkowski operators illustrates the global results. |
| title | Global Leray-Schauder continuation for Fredholm operators |
| topic | Analysis of PDEs Functional Analysis 46T20, 47H11, 47A53, 46G12, 34K18 |
| url | https://arxiv.org/abs/2512.08412 |