Topic
Dini continuity
About: Dini continuity is a research topic. Over the lifetime, 29 publications have been published within this topic receiving 262 citations.
Papers
TL;DR: In this article, the authors studied the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form, with bounded lower order terms, and Dini continuous coefficients, and established uniform spatial Lipschitz estimates for some singular perturbation problems, in the same context.
Abstract: In this article we prove, via monotonicity formulas, interior and boundary gradient esti mates for solutions to second order parabolic equations, in divergence form, with Dini top order coefficients. We then prove uniform Lipschitz estimates for solutions of singular perturbation prob lems, using the previous results, and two phase monotonicity formulas. Introduction. The purpose of this paper is to study the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form, with bounded lower order terms, and Dini continuous coefficients, and also to establish uniform spatial Lipschitz estimates, for some singular perturbation problems, in the same context. In Part 1, we obtain interior and boundary estimates for solutions to these linear equations. As is well-known, the weakest assumption on the modulus of continuity of the top order terms, to obtain interior continuity of the gradient of solutions, is Dini continuity (see Part 1 for the relevant definitions). (See, for instance (G-W) in the elliptic case.) The usual approach to such results is to use appropriate potentials. Here, we use a monotonicity formula approach (Theo rem 1.2.9), which we believe is new. The approach has the advantage that it allows us to show that Dini continuity of the coefficients at a given point yields bound edness of the gradient at the same point. Using our monotonicity formula, the fact that linear functions are "almost" solutions, and a geometric argument from (Cl), we also obtain, under uniform Dini continuity of the coefficients, a uniform modulus of continuity estimate for gradients of solutions (Theorem 1.3.1). The fact that the monotonicity formula at a point depends only on the Dini continuity at that same point enables us to use a simple reflection argument to extend these results up to the boundary (Theorem 1.4.5 and Theorem 1.4.12). We thus obtain a modulus of continuity estimate up to the boundary, for gradients of solutions with smooth Dirichlet data, under uniform Dini continuity of the top order coeffi cients. Once this is established, a compactness argument (Theorem 1.5.10) and a judicious application of Harnack's principle allow us to obtain a Hopf maximum principle (Theorem 1.5.10, Corollary 1.5.13 and Corollary 1.5.16) for our class of equations.
101 citations
TL;DR: In this paper, the existence and uniqueness of mild solutions for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Holder continuous of order larger than 2 3 in the other component were proved.
Abstract: The existence and uniqueness of mild solutions are proved for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Holder continuous of order larger than 2 3 in the other component. In the finite-dimensional case the Dini continuity is further weakened. The main results are applied to solve second order stochastic systems driven by spacetime white noises.
27 citations
Posted Content•
TL;DR: In this paper, an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable is given.
Abstract: In this note we prove an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations.
24 citations
TL;DR: For solutions of the Dirichlet problem for a second-order elliptic equation, an analogue of the Carleson theorem on Lp-estimates was established in this paper.
Abstract: For solutions of the Dirichlet problem for a second-order elliptic equation, we establish an analogue of the Carleson theorem on Lp-estimates. Under the same conditions on the coefficients for which the unique solvability of the considered problem is known, we prove this criterion for the validity of estimate of the solution norm in the space Lp with a measure. We require their Dini continuity on the boundary, but we assume only their measurability and boundedness in the domain under consideration.
22 citations
TL;DR: In this article, the authors prove that the gradient of solutions to obstacle problems are continuous if the gradients of obstacles satisfy a Dini type continuity assumption, and they also consider coefficients and nonhomogeneous data and investigate their regularity conditions to obtain gradient continuity.
Abstract: We prove that the gradients of solutions to obstacle problems are continuous if the gradients of obstacles satisfy a Dini type continuity assumption. We also consider coefficients and nonhomogeneous data and investigate their regularity conditions to obtain gradient continuity, extending to the constrained case results starting with those presented by Mingione (J Eur Math Soc 13:459–486, 2011).
13 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2020 | 1 |
| 2019 | 1 |
| 2018 | 1 |
| 2017 | 3 |
| 2016 | 4 |