{
  "id": "1903.05158",
  "version": "v1",
  "published": "2019-03-12T19:14:00.000Z",
  "updated": "2019-03-12T19:14:00.000Z",
  "title": "Semilinear integro-differential equations, I: odd solutions with respect to the Simons cone",
  "authors": [
    "Juan-Carlos Felipe-Navarro",
    "Tomás Sanz-Perela"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This is the first of two papers concerning saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\\mathbb{R}^{2m}$, where $L_K$ 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 $\\{(x', x'') \\in \\mathbb{R}^m \\times \\mathbb{R}^m \\, : \\, |x'| = |x''|\\}$, and vanish only on this set. By the odd symmetry, $L_K$ coincides with a new operator $L_K^{\\mathcal{O}}$ which acts on functions defined only on one side of the Simons cone, $\\{|x'|>|x''|\\}$, and that vanish on it. This operator $L_K^{\\mathcal{O}}$, which corresponds to reflect a function oddly and then apply $L_K$, has a kernel on $\\{|x'|>|x''|\\}$ 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 $\\tau\\mapsto K(\\sqrt \\tau)$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-12T19:14:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G20",
      "35B06",
      "35B50",
      "35B08"
    ],
    "keywords": [
      "simons cone",
      "semilinear integro-differential equations",
      "saddle-shaped solution",
      "odd solutions",
      "linear elliptic integro-differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}