On general metric measure spaces, we develop a new well-posedness theory for the signed porous medium equation and its fast diffusion counterpart \[ \partial_t u = \mathcal{L}\left(|u|^{m-1}u\right), \qquad m>0, \] where $\mathcal{L}$ is the associated non-positive self-adjoint operator of a symmetric Dirichlet form. The theory does not rely on a Gelfand triple or compact embeddings; instead, it is built upon the extended Dirichlet space $\mathcal{F}_e$ and auxiliary spaces $V^q:=L^q\cap\mathcal{F}_e$, whose uniform convexity plays a key role in the proof. The proof uses only the existence of the Dirichlet form and its extension; no additional regularity of the form or geometric assumptions on the underlying space are needed. Consequently, the results apply to a wide range of metric measure spaces, including non-smooth fractals.
In this paper we establish a necessary and sufficient condition for the solvability of the Dirichlet problem in the unit ball $\mathbb{B}$ for the second order system $\partial_{\underline{x}} f\partial_{\underline{x}}=0$, where $\partial_{\underline{x}}$ stands for the Clifford-algebra valued Dirac operator in $\mathb...
A. M. García, Daniel Alfonso Santiesteban, R. Blaya· 0 citations
We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\u...
Akram Khan, Sagar Gautam, Manil T. Mohan· 0 citations
We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for $s\in(0,1]$ and $n>2s$. Assuming that the manifold is $s$-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introd...
Dorothea-Enrica von Criegern, Gabriele Grillo, D. D. Monticelli· 0 citations
We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty}$ by smooth p...
In this paper we address a basic question that has not been considered in the literature on the parabolic Regularity problem. Let $\Omega$ be the region above the graph of a $\mathrm{Lip}(1,\tfrac12)$ function $\phi$ on $\mathbb R^{n-1}\times\mathbb R$. The corresponding $\dot L^p_{1,1/2}$ Regularity boundary space on...
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 0 citations
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