We establish a Pogorelov type interior $C^2$ estimate for the real twisted Monge-Ampere equation without assuming uniform ellipticity. This equation is fully nonlinear and elliptic, but is neither concave nor a function of the eigenvalues of the Hessian, so the standard techniques for interior second-derivative estimates do not directly apply. Building on the partial Legendre transform formulation of Streets and Warren, we derive a differential identity for the associated positive-definite matrix $W(D^2u)$ and use some linear algebra to control the third-order terms arising in the maximum principle argument. As an application, we prove a Liouville theorem for entire split convex solutions with quadratic growth, extending the rigidity theorem of Streets and Warren by replacing their uniform ellipticity hypothesis with suitable quadratic growth assumptions.
In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouv...
We prove a uniform $L^\infty$-estimate for the parabolic Monge--Amp\`ere equation under uniform H\"ormander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled K\"ahler--Ricci flows on a family of polarized Calabi--Yau manifolds over the punctured disk. An interpo...
We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequal...
We prove strict P\'olya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. Explicit polynomial positivity arguments and angular-l...
We prove an endpoint Pucci maximum principle for domains in $\mathbb R^2$, formulated using a generalization of elementary symmetric polynomials to noninteger orders. As a result, we sharpen an earlier result of Astala, Iwaniec and Martin, and deduce a new Sobolev inequality for $q$-Hessians where $1<q<2$. The fraction...
Gautam Neelakantan Memana, N. Trudinger, Po-Lam Yung· 2 citations
We establish an intrinsic Harnack inequality with an optimal tail term for weak solutions of parabolic fractional $p$-Laplace equations with $p>2$ and $s\in(0,1)$ whose kernels are symmetric, measurable, and comparable to $|x-y|^{-n-sp}$. It constitutes the nonlocal analogue of the intrinsic Harnack inequality of DiBen...
Harsh Prasad· 2 citations
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