Related papers: The $L^p$ regularity problem for parabolic operato…

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Let $\nu=e^{-U}\mu$, where $e^{-U}$ is a sufficiently regular weight and…

In this article, we investigate the H\"{o}lder regularity of the fractional $p$-Laplace equation of the form $(-\Delta_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(\Omega)$. Specifically, we prove that $u\in C^{0,…

In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…

We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\Delta)^s$ on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in…

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in…

We consider divergence form elliptic operators of the form $L=-\dv A(x)\nabla$, defined in $R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic, complex…

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_{\alpha} $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $\mu$ for all $\alpha \in [0,1]$. We study the Dirichlet problem for the…

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…

For a class of non-symmetric non-local L\'evy-type operators $\mathcal{L}^{\kappa}$, which include those of the form $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\,…

We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[ \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb…

Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $\tau$) be a uniformly elliptic operator with boundary conditions on a domain $\Omega$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on…

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form \begin{equation*} \partial_t u^i - \mathrm{div} \big( a(|Du|) Du^i \big)= f^i, \qquad i=1,\dots,N, \end{equation*} in a space-time cylinder…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

In this note, we investigate the regularity of extremal solution $u^*$ for semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=\lambda f(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$…

We obtain optimal regularity in the Sobolev space $W_0^{1,\tau}(\Omega)$ for the unique solution of $$ -\Delta_m u=K(x)u^{-p} \mbox{in} \Omega, \quad u=0\mbox{on}\partial \Omega. $$ Here $\Omega\subset{\mathbb R}^N$ is a smooth and bounded…

We consider the normalized $p$-Poisson problem $$-\Delta^N_p u=f \qquad \text{in}\quad \Omega.$$ The normalized $p$-Laplacian $\Delta_p^{N}u:=|D u|^{2-p}\Delta_p u$ is in non-divergence form and arises for example from stochastic games. We…

We solve the Kato square root problem for parabolic operators of arbitrary order $2m$ whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a…

Let $L$ be a second order elliptic operator on $R^d$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$ with some $p>d$. We assume that $L$ possesses a Lyapunov function, but no local boundedness of…