相关论文: On the existence and H\"older regularity of soluti…

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl…

We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-\Delta)^s(|u|^{m-1}u) = 0…

We study the interior H\"older regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, H\"older regularity is a…

We study the De Giorgi-Moser-Nash estimates of higher-order parabolic equations in divergence form with complex-valued, measurable, bounded, uniformly elliptic (in the sense of G$\mathring{a}$rding inequality) and time-independent…

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…

Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincar\'e inequality, and as a consequence, we deduce a version of the classical Moser…

We develop a discrete counterpart of the De Giorgi-Nash-Moser theory, which provides uniform H\"older-norm bounds on continuous piecewise affine finite element approximations of second-order linear elliptic problems of the form $-\nabla…

In this paper, we establish the De Giorgi-Nash-Moser theory for a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, with the fractional $p$-Laplacian operator in Grushin-type spaces…

We establish the interior and boundary H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-\Delta_p u=0,\quad p>1. \] The proof…

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove…

We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are…

We establish the H\"older continuity of bounded nonnegative weak solutions to \begin{align*} \big(\Phi^{-1}(w)\big)_t=\Delta w+\nabla\cdot\big(a(x,t)\Phi^{-1}(w)\big)+b\big(x,t,\Phi^{-1}(w)\big), \end{align*} with convex $\Phi\in…

We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, the H\"older regularity and the Schauder estimates to a class of degenerate parabolic equations of non-divergence form in all dimensions \begin{equation}…

We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The…

In this paper, using the De Giorgi-Nash-Moser method, we obtain an interior H\"older continuity of weak solutions to nonlocal $p$-Laplacian type Schr\"odinger equations given by an integro-differential operator ${\rm L}^p_K$ ($p>1$) as…

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$ driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further…

We prove H\"older regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved.…

We investigate existence and regularity of weak solutions of a 1-dimensional parabolic differential equation with a non-constant H\"older diffusion coefficient and a rough forcing term. Such an equation appears in studying the 1-dimensional…

We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate…

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…