We consider the Hamiltonian stationary equation for all phases in dimension two. We show that solutions that are $C^{1,1}$ will be smooth and we also derive a $C^{2,alpha}$ estimate for it.
Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Holder semi-norms not with respect to all, but only with respect to some of the independent
variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.
In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obt
ained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even $m$ it depends on the integrability index $p$. The sharpness of the new-found coercivity condition is demonstrated by an example.
In this note, we use the non-homogeneous Poisson stochastic process to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The method is probability. We generalize the result of Krylov-Priola [7].
We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some app
lications of this regularity result, especially in the calculus of variations for the area functional.
Arunima Bhattacharya
,Micah Warren
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(2018)
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"Interior Schauder estimates for the fourth order Hamiltonian stationary equation in two dimensions"
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Arunima Bhattacharya
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