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Register a new userOn $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II
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We extend and improve the results in cite{DK16}: showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form equations. We extend the weak type-(1, 1) estimates in cite{DK16} and cite{Es94} up to the boundary and derive a Harnack inequality for non-negative adjoint solutions to non-divergence form elliptic equations, when the leading coefficients have Dini mean oscillation.
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We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the $L^1$-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Brezis. We also prove a weak type-$(1,1)$ estimate under a stronger assumption on the modulus of continuity. The corresponding results for non-divergence form equations are also established.
We show that weak solutions to parabolic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form parabolic operators and their adjoint operators. Under similar conditions, we also establish a Harnack inequality for nonnegative adjoint solutions, together with upper and lower Gaussian bounds for the global fundamental solution.
In this paper, we show $C^{2,alpha}$ interior estimates for viscosity solutions of fully non-linear, uniformly elliptic equations, which are close to linear equations and we also compute an explicit bound for the closeness.
The art of analysis involves the subtle combination of approximation, inequalities, and geometric intuition as well as being able to work at different scales. With this subtlety in mind, we present this paper in a manner designed for wide accessibility for both advanced undergraduate students and graduate students. The main results include a singular integral for measuring the level sets of a $C^{1,1}$ function mapping from $mathbb{R}^n$ to $mathbb{R}$, that is, one whose derivative is Lipschitz continuous. We extend this to measure embedded submanifolds in $mathbb{R}^2$ that are merely $C^1$ using the distance function and provide an example showing that the measure does not hold for general rectifiable boundaries.
In this paper we obtain $C^{1,theta}$-estimates on the distance of inertial manifolds for dynamical systems generated by evolutionary parabolic type equations. We consider the situation where the systems are defined in different phase spaces and we estimate the distance in terms of the distance of the resolvent operators of the corresponding elliptic operators and the distance of the nonlinearities of the equations.
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