Related papers: Green's function for second order elliptic equatio…
We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse…
We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a…
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…
We investigate boundary estimates for elliptic operators with stationary random coefficients exhibiting integrable correlations, arising from stochastic homogenization theory. As practical applications, we establish decay estimates for…
We study inhomogeneous nonlinear second-order differential equations in one dimension. The inhomogeneities can be point sources or continuous source distributions. We consider second order differential equations of type $\phi''(x) +…
We consider gradient estimates for $H^1$ solutions of linear elliptic systems in divergence form $\partial_\alpha(A_{ij}^{\alpha\beta} \partial_\beta u^j) = 0$. It is known that the Dini continuity of coefficient matrix $A =…
We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to…
General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic…
We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \mathcal{L}_K=-\text{div}(K(x)\nabla ) $ on a bounded smooth domain $ \Omega\subseteq\mathbb{R}^n (n\geq 2) $ with zero Dirichlet boundary condition, where $…
We study a class of second-order elliptic equations of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. We establish optimal gradient estimates by showing the explicit…
Green's functions for Neumann boundary conditions have been considered in Math Physics and Electromagnetism textbooks, but special constraints and other properties required for Neumann boundary conditions have generally not been noticed or…
Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged…
We look at estimates for the Green's function of time-fractional evolution equations of the form $D^{\nu}_{0+*} u = Lu$, where $D^{\nu}_{0+*}$ is a Caputo-type time-fractional derivative, depending on a L\'evy kernel $\nu$ with variable…
In Boundary Element Method, Green's function with no boundary conditions is used for solving Laplace's equation with Dirichlet boundary condition. To determine the gradient of solution on the boundary, we need to solve the boundary integral…
The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have…
In this article we use linear algebra to improve the computational time for the obtaining of Green's functions of linear differential equations with reflection (DER). This is achieved by decomposing both the `reduced' equation (the ODE…
We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second order equation in divergence form with discontinuous coefficient. Our concern is to estimate the solutions with explicit constants,…
In this paper we develop a way of obtaining Green's functions for partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat…
It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the…
This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative…