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• W2922122562 endingPage "108309" @default.
• W2922122562 startingPage "108309" @default.
• W2922122562 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 $taumapsto K(sqrt tau)$ 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." @default.
• W2922122562 created "2019-03-22" @default.
• W2922122562 creator A5004763893 @default.
• W2922122562 creator A5057231886 @default.
• W2922122562 date "2020-01-01" @default.
• W2922122562 modified "2023-09-25" @default.
• W2922122562 title "Semilinear integro-differential equations, I: Odd solutions with respect to the Simons cone" @default.
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• W2922122562 doi "https://doi.org/10.1016/j.jfa.2019.108309" @default.
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