Semilinear integro-differential equations, I: Odd solutions with respect to the Simons cone
This is the first of two papers concerning saddle-shaped solutions to the semilinear equation in , where is a linear elliptic integro-differential operator and f is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone, and vanish only on this set. By the...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/530712 |
| Acceso en línea: | http://hdl.handle.net/2072/530712 |
| Access Level: | acceso abierto |
| Palabra clave: | Matemàtiques 51 |
| Sumario: | This is the first of two papers concerning saddle-shaped solutions to the semilinear equation in , where is a linear elliptic integro-differential operator and f is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone, and vanish only on this set. By the odd symmetry, coincides with a new operator which acts on functions defined only on one side of the Simons cone, , and that vanish on it. This operator , which corresponds to reflect a function oddly and then apply , has a kernel on which is different from K. In this first paper, we characterize the kernels K for which the new kernel is positive and therefore one can develop a theory on the saddle-shaped solution. The necessary and sufficient condition for this turns out to be that K is radially symmetric and is a strictly convex function. Assuming this, we prove an energy estimate for doubly radial odd minimizers and the existence of saddle-shaped solution. In a subsequent article, part II, further qualitative properties of saddle-shaped solutions will be established, such as their asymptotic behavior, a maximum principle for the linearized operator, and their uniqueness. |
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