2,272 research outputs found
The prescribed mean curvature equation in weakly regular domains
We show that the characterization of existence and uniqueness up to vertical
translations of solutions to the prescribed mean curvature equation, originally
proved by Giusti in the smooth case, holds true for domains satisfying very
mild regularity assumptions. Our results apply in particular to the
non-parametric solutions of the capillary problem for perfectly wetting fluids
in zero gravity. Among the essential tools used in the proofs, we mention a
\textit{generalized Gauss-Green theorem} based on the construction of the weak
normal trace of a vector field with bounded divergence, in the spirit of
classical results due to Anzellotti, and a \textit{weak Young's law} for
-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector
fields have been now extended and moved in a self-contained paper available
at: arXiv:1708.0139
Surface parameterization over regular domains
Surface parameterization has been widely studied and it has been playing a critical role in many geometric processing tasks in graphics, computer-aided design, visualization, vision, physical simulation and etc. Regular domains, such as polycubes, are favored due to their structural regularity and geometric simplicity. This thesis focuses on studying the surface parameterization over regular domains, i.e. polycubes, and develops effective computation algorithms. Firstly, the motivation for surface parameterization and polycube mapping is introduced. Secondly, we briefly review existing surface parameterization techniques, especially for extensively studied parameterization algorithms for topological disk surfaces and parameterizations over regular domains for closed surfaces. Then we propose a polycube parameterization algorithm for closed surfaces with general topology. We develop an efficient optimization framework to minimize the angle and area distortion of the mapping. Its applications on surface meshing, inter-shape morphing and volumetric polycube mapping are also discussed
On a decomposition of regular domains into John domains with uniform constants
We derive a decomposition result for regular, two-dimensional domains into
John domains with uniform constants. We prove that for every simply connected
domain with -boundary there is a corresponding
partition with such
that each component is a John domain with a John constant only depending on
. The result implies that many inequalities in Sobolev spaces such as
Poincar\'e's or Korn's inequality hold on the partition of for uniform
constants, which are independent of
Spectral estimates of the -Laplace Neumann operator in conformal regular domains
In this paper we study spectral estimates of the -Laplace Neumann operator
in conformal regular domains . This study is based on
(weighted) Poincar\'e-Sobolev inequalities. The main technical tool is the
composition operators theory in relation with the Brennan's conjecture. We
prove that if the Brennan's conjecture holds then for any and
the weighted -Poincare-Sobolev inequality holds with
the constant depending on the conformal geometry of . As a consequence
we obtain classical Poincare-Sobolev inequalities and spectral estimates for
the first nontrivial eigenvalue of the -Laplace Neumann operator for
conformal regular domains.Comment: 15 page
- …