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Learning Seminar on PDE and Applications

Critical Point Sets of Solutions in Elliptic Homogenization

Fanghua Lin (Courant Institute, NYU)

Location:  Hill 525
Date & time: Thursday, 28 September 2023 at 2:00PM - 3:20PM

Abstract: The quantitative uniqueness and the geometric measure estimates for the nodal and critical point sets of solutions of second order elliptic equations depend crucially on the bound of the associated Almgren's frequency function. The latter is possible (only) when the leading coefficients of equations are uniformly Lipschitz. One does not have this uniform Lipschitz continuity for coefficients of equations in elliptic homogenization. Instead, by using quantitative homogenization, successive harmonic approximation and suitable L^2-renormalization, we shall see how one can get a uniform estimate (independent of a small parameter characterizing the nature of homogenization) of co-dimension two Hausdorff measure as well as the Minkowski content of the critical point sets. A key element is an estimate of "turning" for the projection of a non-constant solution onto the subspace of spherical harmonics of order N, when the doubling index of solution on annular regions is trapped near N.

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