Concavity of solutions to elliptic equations on the sphere

Abstract: An important question in PDE is when a solution to an elliptic equation is concave. This has been of interest with respect to the spectrum of linear equations as well as in nonlinear problems. An old technique going back to works of Korevaar, Kennington and Kawohl is to study a certain two-point function on a Euclidean domain to prove a so-called concavity maximum principle with the help of a first and second derivative test. To our knowledge, so far this technique has never been transferred to other ambient spaces, as the nonlinearity of a general ambient space introduces geometric terms into the classical calculation, which in general do not carry a sign. In this talk we have a look at this situation on the unit sphere. We prove a concavity maximum principle for a broad class of degenerate elliptic equations via a careful analysis of the spherical Jacobi fields and their derivatives. In turn we obtain concavity of solutions to this class of equations. This is joint work with Mat Langford, University of Tennessee Knoxville.

Mathematics

Audience: researchers in the topic

Online Seminar "Geometric Analysis"

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