Related papers: The Kato square root problem for weighted paraboli…

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of…

By using some deep tools from microlocal analysis, the authors of the papers (Ann. of Math., 165 (2007), 567--591, J. Amer. Math. Soc., 23 (2010), 655--691; Invent. Math., 178 (2009), 119--171; Duke Math. J., 158(2011), 83--120) have…

This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of…

We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing…

In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO…

This paper presents a mathematical analysis of a doubly degenerate parabolic equation and its application to the Richards equation using a bounded auxiliary variable. We establish the existence of weak solutions using semi-implicit time…

In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial $A_2$-weight. We prove that having a generalized…

We prove the $W^{1,2}_p$-estimate and solvability for the Dirichlet problem of second-order parabolic equations in simple convex polytopes with time irregular coefficients, when $p\in (1,2]$. We also consider the corresponding Neumann…

We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let $L_w$ be a degenerate elliptic operator with degeneracy given by a fixed weight $w\in…

For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the…

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…

The aim of this work is to prove existence and uniqueness results for a doubly nonlinear elliptic problem that is essential for solving the associated parabolic problem using Rothe's method (discretizing time). We work under very weak…

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in…

We derive weighted Sobolev-Poincar\'e type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class,…

We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity…

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator $L$ defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on…

We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \partial_t^\alpha u - Lu + \lambda u= f \quad \mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^\alpha u$ is the Caputo…

We study elliptic and parabolic problems governed by the singular elliptic operators $$ \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_x\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be H\"older or Dini continuous in the time variable and all but one spatial variables. This…

This paper studies a priori and regularity estimates of Evans-Krylov type in H\"older spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of…