Dini Continuous Coefficients Research Articles

Formula omitted]-partial regularity for sub-elliptic systems with Dini continuous coefficients in Carnot groups

In this paper, we consider nonlinear sub-elliptic systems with Dini continuous coefficients for the case 1<m<2 in Carnot groups. Based on a generalization of the technique of A-harmonic approximation introduced by Duzaar–Grotowski–Kronz, and a Sobolev–Poincaré type inequality established in Carnot groups, a C1-partial regularity result for weak solutions of sub-elliptic systems with natural growth terms is established. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients it yields directly the optimal Hölder exponent for horizontal gradients of weak solutions on its regular set.

Partial interior regularity for sub-elliptic systems with Dini continuous coefficients in Carnot groups: the sub-quadratic controllable case

We consider nonlinear sub-elliptic systems with Dini continuous coefficients for the case 1< m<2 in Carnot groups and prove a C^{1}-partial regularity result for weak solutions under the controllable growth conditions. Our method of proof for sub-elliptic systems is based on a generalization of the technique of mathcal {A}-harmonic approximation. It is interesting to point out that our result is optimal in the sense that in the case of Hölder continuous coefficients we get directly the optimal Hölder exponent on its regular set.

Dirichlet problem for parabolic systems with Dini continuous coefficients on the plane

The Dirichlet problem for a one-dimensional (with respect to x) second-order parabolic system with Dini continuous coefficients is considered in an x-semibounded domain with a nonsmooth lateral boundary from the Dini–Holder class. The classical solvability of the problem is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.

On the C 1 regularity of solutions to divergence form elliptic systems with dini-continuous coefficients

We prove $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients

Optimal partial regularity for nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups

This paper is concerned with the interior regularity for nonlinear sub-elliptic systems with Dini continuous coefficients under superquadratic controllable growth conditions in Carnot groups. We adapt the technique of the $\mathcal{A}$ -harmonic approximation to the case of sub-elliptic systems in divergence form, and we show a partial regularity result for weak solutions. In particular, our result is optimal in the sense that in the case of Holder continuous coefficients we obtain directly the optimal Holder exponent for the horizontal gradient of weak solutions on its regular set.

Optimal partial regularity for sub-elliptic systems with Dini continuous coefficients under the superquadratic natural growth

The paper is concerned with nonlinear sub-elliptic systems with Dini continuous coefficients under super-quadratic (m>2) natural growth conditions in the Heisenberg group. A general criterion for weak solutions to be regular in the neighborhood of a given point is established by a generalization of method of A-harmonic approximation. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients we obtain directly the optimal Hölder exponent for horizontal gradient of weak solutions on its regular set.

Partial regularity for sub-elliptic systems with Dini continuous coefficients involving natural growth terms in the Heisenberg group

We consider nonlinear sub-elliptic systems with Dini continuous coefficients in divergence form in the Heisenberg group. Based on a generalization of the method of A-harmonic approximation introduced by Duzaar and Steffen, partial regularity of weak solutions for sub-elliptic systems under natural growth conditions is established. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients we obtain directly the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.

Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups

This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg groupℍn. Based on a generalization of the technique of𝒜-harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.

Regularity for nonlinear elliptic systems with dini coefficients under natural growth condition for the case: 1

In this article, we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1 <m < 2. All estimates in the case of m ≥ 2 is no longer suitable, and we can't obtain the Caccioppoli's second inequality by using these techniques developed in the case of m ≥ 2. But the Caccioppoli's second inequality is the key to use A-harmonic approximation method. Thus, we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition. And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.

Optimal partial regularity of second order nonlinear elliptic systems with Dini continuous coefficients for the superquadratic case

We consider the regularity for weak solutions of second order nonlinear elliptic systems with Dini continuous coefficients for the superquadratic case under natural growth condition, and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal Hölder exponent for the derivative of the weak solutions on the regular set.

Interior partial regularity for nonlinear elliptic systems with Dini continuous coefficients for the case: [formula omitted

In this article, we consider the interior regularity for weak solutions of nonlinear elliptic systems with divergence type and Dini continuous coefficients under subquadratic growth conditions. We adopt another A -harmonic approximation technique and prove the interior partial regularity of weak solutions. The proof yields directly the optimal Hölder exponent for the derivative of the weak solutions on the regular set.

Two-sided estimates on Dirichlet heat kernels for time-dependent parabolic operators with singular drifts in [formula omitted]-domains

In this paper, we establish sharp two-sided estimates for the Dirichlet heat kernels of a large class of time-dependent parabolic operators with singular drifts in C 1 , α -domain in R d , where d ⩾ 1 and α ∈ ( 0 , 1 ] . Our operator is L + μ ⋅ ∇ x , where L is a time-dependent uniformly elliptic divergent operator with Dini continuous coefficients and μ is a signed measure on ( 0 , ∞ ) × R d belonging to parabolic Kato class. Along the way, a gradient estimate is also established. Our method employs a combination of partial differential equations and perturbation techniques.

Regularity for a Class of Elliptic Operators with Dini Continuous Coefficients

Regularity for a Class of Elliptic Operators with Dini Continuous Coefficients

Optimal interior partial regularity for nonlinear elliptic systems with Dini continuous coefficient

In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal Hölder exponent for the derivative of a weak solution on its regular set.

Regularity for a Class of Elliptic Operators with Dini Continuous Coefficients

We obtain regularity results for solutions to Pu=f when P is a kth order elliptic differential operator with the property that both P and Pt have coefficients that are continuous, with a modulus of continuity satisfying a Dini-type condition. Operators of this form arise in the study of regularity of functions that are known to be regular along the leaves of several foliations, such as arise in Anosov systems. The results here complement some previous results of the author and J. Rauch.

On a conjecture of J. Serrin

In 1964 J. Serrin proposed the following conjecture. Let u \in W_{\mathrm{loc}}^{1,1}(Ω) be a weak solution of the second order elliptic equation (1) below, in divergence form, with Hölder continuous coefficients a_{ij}(x) ; then u is a “classical” solution. We announce a solution of this conjecture assuming only u \in BV_{\mathrm{loc}}(Ω) and Dini continuous coefficients.

Nonlinear elliptic systems with Dini continuous coefficients

We consider nonlinear elliptic systems of divergence type with Dini continous coefficients and prove a partial regularity result for weak solutions. Our method of proof is based on a generalization of the technique of harmonic approximation. Our result is optimal in the sense that in the case of Holder continous coefficients we establish the optimal Holder exponent for the derivative of the weak solution on its regular set.