Related papers: Pointwise estimates for degenerate Kolmogorov equa…

By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup $P_t$ generated by the Gruschin type operator on $\R^{m}\times \R^{d}:$ $$L (x,y)=\ff 1 2 \bigg\{\sum_{i=1}^m \pp_{x_i}^2…

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr\"odinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such…

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of…

We consider degenerate Monge-Ampere equations of the type $$\det D^2 u= f \quad \{in $\Om$}, \quad \quad f \sim \, d_{\p \Om}^\alpha \quad \{near $\p \Om$,}$$ where $d_{\p \Om}$ represents the distance to the boundary of the domain $\Om$…

We prove sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient…

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…

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions…

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after…

In this paper we investigate the validity of first and second order $L^{p}$ estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present $L^{p}$ estimates of the gradient under…

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity…

We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on…

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…

Let $u_\e$ be a solution to the system $$ \mathrm{div}(A_\e(x) \nabla u_{\e}(x))=0 \text{\ in} D, \qquad u_{\e}(x)=g(x,x/\e) \text{\ on}\partial D, $$ where $D \subset \R^d $ ($d \geq 2$), is a smooth uniformly convex domain, and $g$ is…

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,…

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 derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of…

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one…

In this paper, we establish a Liouville type theorem for the homogeneous dual fractional parabolic equation \begin{equation} \partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = 0\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R} . \end{equation} where…

We show the existence and uniqueness of fundamental solution operators to Kolmo\-gorov-Fokker-Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that…