In this article, we study the weighted homogeneous Sobolev--Morrey spaces on domains in $\mathbb{R}^n$ with higher co-dimensional boundaries. Precisely, we systematically establish a real-variable theory of these spaces, including completeness, embedding theorems, Riesz potential characterizations, continuity, trace and extension theorems, and complex interpolation. Applying the boundedness of the trace and the extension operators, we obtain sharp weighted a priori estimates for solutions to the Dirichlet problem of divergence-form degenerate second-order elliptic equations on such domains in weighted Lebesgue spaces. The absence of a boundary manifold structure of these domains poses some essential difficulties, which are overcome by using some tools, such as the intrinsic properties of distance weights and the geometric structure of domains, different from those available in Lipschitz domains.
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.· 2 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...
We establish the regularity properties of adjoint solutions to elliptic equations in double divergence form with variable coefficients and rough forcing terms. More precisely, we consider Dirichlet problems associated with the formal adjoint of second-order elliptic operators in non-divergence form, assuming that the...
G. Di Fazio, Dennys Sberna· Zeitschrift für Analysis und...· 1 citation
Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This div...
Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically re...
H. Egger, Matthias Schlottbom· 1 citation
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