Related papers: Gradient Estimates for Parabolic Systems from Comp…

For an initial-boundary value problem for a parabolic equation in the spatial variable $x=(x_1,.., x_n)$ and time $t$, we consider an inverse problem of determining a coefficient which is independent of one spatial component $x_n$ by extra…

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric,…

We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. We consider nonsmooth initial data and a backward…

We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $p$-Laplace equation $u_t-\Delta_p u=\mu$ with $p\in (1,2)$. The case…

Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions…

We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the…

We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible…

The aim of this thesis is to derive new gradient estimates for parabolic equations. The gradient estimates found are independent of the regularity of the initial data. This allows us to prove the existence of solutions to problems that have…

We establish derivative estimates of solution of elliptic system in narrow regions.

We establish the Caccioppoli inequality, a reverse H\"older inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order $2m$ with certain lower order…

We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal…

This paper is devoted to the weighted estimates and the solvability of time-fractional parabolic equations. The leading coefficients \(a^{ij}(t,x)\) are assumed to have small mean oscillations in \((t,x)\) locally, in both non-divergence…

We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results…

We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to…

We establish sharp higher-order H\"older regularity estimates up to the boundary for solutions to equations of the form $\partial_t u-Lu=f(t,x)$ in $I\times\Omega$ where $I\subset\mathbb{R}$, $\Omega\subset\mathbb{R}^n$ and $f$ is H\"older…

We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate parabolic and elliptic equations admit estimates of spatial derivatives up to any given order…

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with VMO$_x$…

We develop a new real-variable method for weighted $L^p$ estimates. The method is applied to the study of weighted $W^{1, 2}$ estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded…

In this paper, we complete the analysis initiated in [AFV24] establishing some higher order $C^{k+2,\alpha}$ Schauder estimates ($k \in \mathbb{N}$) for a a class of parabolic equations with weights that are degenerate/singular on a…

In this article we consider variable coefficient, time-dependent wave equations. Using phase space methods we construct outgoing parametrices and prove Strichartz-type estimates globally in time. This is done in the context of C^2 metrics…