Gradient estimate for a special Lagrangian type PDE

Ravi Shankar (Princeton University)
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Wednesday PDL C401

We revisit gradient estimates for fully nonlinear elliptic PDEs without uniform ellipticity.  In joint work with Arunima Bhattacharya and Connor Mooney, a gradient estimate was established for a special Lagrangian PDE with variable phase (the Lagrangian mean curvature equation).  Micah Warren and Yu Yuan obtained a gradient estimate, in 2009 and 2010, for the constant phase equation, if the phase is sufficiently large: critical or supercritical.  Because the ellipticity degenerates at the critical level, recent work has required a strictly supercritical condition on the phase.  Our result includes the borderline, critical case, if the phase is merely C^2, or more generally satisfies a certain differential inequality which ensures flatness at the critical values.  This provides the missing link for the Dirichlet problem for such phases.  

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