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Tangent measure and blow-up methods are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly nonsymmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains , :
We extend the results of Kenig, Preiss, and Toro (J. Amer. Math. Soc. 22:3 (2009), 771–796) by showing mutual absolute continuity of interior and exterior elliptic measures for any domains implies the tangent measures are a.e. flat and the elliptic measures have dimension .
We generalize the work of Kenig and Toro (J. Reine Agnew. Math. 596 (2006), 1–44) and show that equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always supported on the zero sets of elliptic polynomials.
In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower -Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to -Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell (Trans. Amer. Math. Soc. 369:8 (2017), 5711–5745).
Chae and Wolf recently constructed discretely self-similar solutions to the Navier–Stokes equations for any discretely self-similar data in . Their solutions are in the class of local Leray solutions with projected pressure and satisfy the “local energy inequality with projected pressure”. In this note, for the same class of initial data, we construct discretely self-similar suitable weak solutions to the Navier–Stokes equations that satisfy the classical local energy inequality of Scheffer and Caffarelli–Kohn–Nirenberg. We also obtain an explicit formula for the pressure in terms of the velocity. Our argument involves a new purely local energy estimate for discretely self-similar solutions with data in and an approximation of divergence-free, discretely self-similar vector fields in by divergence-free, discretely self-similar elements of .
We study the asymptotic behavior at rational directions of the effective boundary condition in periodic homogenization of oscillating Dirichlet data. We establish a characterization for the directional limits at a rational direction in terms of a relatively simple two-dimensional boundary layer problem for the homogenized operator. Using this characterization we show continuity of the effective boundary condition for divergence form linear systems, and for divergence form nonlinear equations we give an example of discontinuity.
We are concerned with the dynamics of one-fold symmetric patches for the two-dimensional aggregation equation associated to the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows us to analyze the concentration phenomenon of the aggregation patches near the blow-up time. In particular, we prove that the patch collapses to a collection of disjoint segments and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.
We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler–Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau–Tian–Donaldson conjecture and as a corollary we obtain an example of a coupled Kähler–Einstein metric on a manifold which does not admit Kähler–Einstein metrics. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton-type equations on toric Fano manifolds.
We prove that the Dirichlet problem for degenerate elliptic equations in the upper half-space is solvable when and the boundary data is in for some . The coefficient matrix is only assumed to be measurable, real-valued and -independent with a degenerate bound and ellipticity controlled by an -weight . It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is in with respect to the -weighted Lebesgue measure on . The Carleson measure estimate allows us to avoid applying the method of -approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.
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