No Arabic abstract
We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $sigmain (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
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].
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.
This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perrons method. If the equation is uniformly elliptic in the sense of cite{SS}, we prove the discontinuous viscosity solution is Holder continuous and thus it is a viscosity solution.
This paper proves Holder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Holder continuous.
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Gauduchon which extends the Calabi conjecture; this was one of the original motivations to this work. We were also motivated by the fact that there had been increasing interests in fully nonlinear pdes from complex geometry in recent years, and aimed to develop general methods to cover as wide a class of equations as possible.