Related papers: Pointwise two-scale expansion for parabolic equati…
In the whole space $R^d$ ($d\ge 2$), we study homogenization of a divergence-form matrix elliptic operator $L_\varepsilon$ of an arbitrary even order larger than 2 with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is…
We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb…
We find two series expansions for Legendre's second incomplete elliptic integral $E(\lambda, k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(\lambda, k)$ plane.…
For a parabolic equation associated to a uniformly elliptic operator, we obtain a $W^{3, \varepsilon}$ estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. In [3] rate of convergence results in homogenization and estimates on the difference between the averaged…
We analyze the convergence of the exponential Lie and exponential Strang splitting applied to inhomogeneous second-order parabolic equations with Dirichlet boundary conditions. A recent result on the smoothing properties of these methods…
For any $\alpha \in (0,1)$, we construct an example of a solution to a parabolic equation with measurable coefficients in two space dimensions which has an isolated singularity and is not better that $C^\alpha$. We prove that there exists…
We develop a quantitative theory of stochastic homogenization in the more general framework of differential forms. Inspired by recent progress in the uniformly elliptic setting, the analysis relies on the study of certain subadditive…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…
We consider the pointwise in space Lp-type regularity for elliptic and parabolic equations of order m in Rn. We provide pointwise Schauder estimates for the general range of Lp exponents, extending previous results from p > n/m to 1 < p <…
We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel [Arxiv preprint 1410.4483, 2014] on the…
We establish a new sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. For real-analytic leading coefficients, we prove a localised estimate of the nodal set, at a given time-level, that…
In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\varepsilon = (\varepsilon_1, \varepsilon_2, \cdots) \in (0,1)^\infty$, with…
We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighbourhood of the logarithmic…
In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse H\"{o}lder's inequalities…
An asymptotic investigation of monochromatic electromagnetic fields in a layered periodic medium is carried out under the assumption that the wave frequency is close to the frequency of a stationary point of the dispersion surface. We find…
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour…