Dini Continuous Coefficients Research Articles

Jump relation for the conormal derivative of parabolic single layer potential

We consider a single layer potential generated by the fundamental solution of a second-order parabolic equation with Dini continuous coefficients. The potential is considered in a semibounded spatial domain with non-smooth lateral boundary from the Dini–Hölder class. We establish a jump relation for the conormal derivative of this potential at the lateral boundary of the domain.

Regular Fundamental Solution to Parabolic Equation with Dini Continuous Coefficients in Many Spatial Variables

Regular Fundamental Solution to Parabolic Equation with Dini Continuous Coefficients in Many Spatial Variables

Volume Potential and Initial–Boundary Value Problems for Parabolic Systems with Dini Continuous Coefficients in the Plane

Volume Potential and Initial–Boundary Value Problems for Parabolic Systems with Dini Continuous Coefficients in the Plane

Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients

Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients

Uniqueness of Solutions to Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in a Semibounded Domain on the Plane

The first and second initial-boundary value problems for second-order parabolic systems with coefficients satisfying the Dini condition in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps are considered. Theorems on the uniqueness of classical solutions of these problems in the class of functions that are continuous and bounded together with their first spatial derivatives in the closure of this domain are proved.

Uniqueness of Solutions to Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in a Semibounded Domain on the Plane

Uniqueness of Solutions to Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in a Semibounded Domain on the Plane

On the Fundamental Solution of a Parabolic Equation with Dini Continuous Coefficients

We consider a parabolic equation in one spatial variable with Dini continuous coefficients. For this equation, the existence of a classical fundamental solution is proved and estimates are given. The condition on the nature of the continuity of the leading coefficient of the equation for the existence of a fundamental solution is sharp.

On the Fundamental Solution of a Parabolic Equation with Dini Continuous Coefficients

On the Fundamental Solution of a Parabolic Equation with Dini Continuous Coefficients

Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded C1,1 domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

Gradient regularity for non-autonomous functionals with Dini or non-Dini continuous coefficients

We prove \(C^1\) regularity for local vectorial minimizers of the non-autonomous functional$$w\in W^{1,1}_{\rm loc}(\Omega;\mathbb{R}^N)\mapsto \int_{\Omega}b(x)\big[|Dw|^p+a(x)|Dw|^p\log(e+|Dw|)\big] \,dx\,,$$with \(\Omega\) open subset of \(\mathbb{R}^n\), \(n\geq2\) , \(p>1\),\(0\leq a(\cdot)\leq \|a\|_{L^{\infty}(\Omega)}<\infty\), and\(0<\nu\leq b(\cdot)\leq L\). The result is valid provided that the function \(a(\cdot)\)is \(\log\)-Dini continuous and that the coefficient $b(\cdot)$ is Dini continuous orit is weakly differentiable and its gradient locally belongs to the Lorentz space\(L^{n,1}(\Omega;\mathbb{R}^n)\).

The Cauchy Problem for a Multi-Dimensional Parabolic Equation with Dini-Continuous Coefficients

The Cauchy Problem for a Multi-Dimensional Parabolic Equation with Dini-Continuous Coefficients

Uniqueness of Solutions of Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in Domains on the Plane

Uniqueness of Solutions of Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in Domains on the Plane

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

<p style='text-indent:20px;'>We study the regularity properties of the second order linear operator in <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb {R}}}^{N+1} $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k = 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum\limits_{j,k = 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $\end{document}</tex-math></inline-formula> are real valued matrices with constant coefficients, with <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> symmetric and strictly positive. We prove that, if the operator <inline-formula><tex-math id="M4">\begin{document}$ {\mathscr{L}} $\end{document}</tex-math></inline-formula> satisfies Hörmander's hypoellipticity condition, and <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a Dini continuous function, then the second order derivatives of the solution <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> to the equation <inline-formula><tex-math id="M7">\begin{document}$ {\mathscr{L}} u = f $\end{document}</tex-math></inline-formula> are Dini continuous functions as well. We also consider the case of Dini continuous coefficients <inline-formula><tex-math id="M8">\begin{document}$ a_{jk} $\end{document}</tex-math></inline-formula>'s. A key step in our proof is a Taylor formula for classical solutions to <inline-formula><tex-math id="M9">\begin{document}$ {\mathscr{L}} u = f $\end{document}</tex-math></inline-formula> that we establish under minimal regularity assumptions on <inline-formula><tex-math id="M10">\begin{document}$ u $\end{document}</tex-math></inline-formula>.</p>

Contact problem for a second-order parabolic equation with Dini-continuous coefficients

Contact problem for a second-order parabolic equation with Dini-continuous coefficients

Mixed Problems for Plane Parabolic Systems and Boundary Integral Equations

We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coefficients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we find a classical solution of this problem. We investigate the smoothness of the solution as well.

Regularity of Solution to the Cauchy Problem for Parabolic Equation in the Dini Space

We study the smoothness of a solution to the Cauchy problem for a one-dimensional parabolic equation with Dini–continuous coefficients in the Dini space. We obtain estimates for the volume parabolic potential and its spatial first order derivative in the Dini space.

Dirichlet problem for parabolic systems with Dini continuous coefficients

ABSTRACT We consider the first initial boundary value problem for spatially one-dimensional parabolic second-order systems with Dini continuous coefficients in a semibounded domain with nonsmooth lateral boundary. We only require that the right-hand side of the boundary condition has continuous derivative of order 1/2 vanishing at t = 0. By the boundary integral equations method we construct a classical solution of this problem. The smoothness of the solution is studied.

Mixed Problem for a Parabolic System on a Plane and Boundary Integral Equations

We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coefficients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we find a classical solution of this problem. We investigate the smoothness of solution as well.

Smoothness in the Dini Space of a Single Layer Potential for a Parabolic System in the Plane

We prove estimates in the Dini space for the second order spatial derivative of a vector parabolic single layer potential. We obtain estimates for higher order derivatives of solutions to initial-boundary-value problems for one-dimensional parabolic systems with Dini-continuous coefficients in a curvilinear domain with nonsmooth lateral boundary.

Bitsadze–Samarskii problem for parabolic systems with Dini continuous coefficients

ABSTRACTWe consider a nonlocal initial-boundary value problem of the Bitsadze–Samarskii type for spatially one-dimensional parabolic second-order systems with Dini continuous coefficients in a domain with nonsmooth lateral boundaries. By the boundary integral equations method we construct a classical solution of this problem. The smoothness of the solution is studied.