Real linear operators arise in a range of applications of mathematical physics. In this paper, basic properties of real linear operators are studied and their spectral theory is developed. Suitable extensions of classical operator theoretic concepts are introduced. Providing a concrete class, real linear multiplication operators are investigated and, motivated by the Beltrami equation, related ...

For a smooth, compact Riemannian manifold (M, g) of dimension N ≥ 3, we are interested in the critical equation ∆gu+ ( N − 2 4(N − 1) Sg +εh ) u = u N+2 N−2 in M , u > 0 in M , where ∆g is the Laplace–Beltrami operator, Sg is the Scalar curvature of (M, g), h ∈ C (M), and ε is a small parameter.

We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...

The manipulation of surface homeomorphisms is an important aspect in 3D modeling and surface processing. Every homeomorphic surface map can be considered as a quasiconformal map, with its local non-conformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasiconformal maps are of great interest in mathematical study and real applications. Efficient and ac...

The Beltrami image flow is an effective nonlinear filter, often used in color image processing. It was shown to be closely related to the median, total variation, and bilateral filters. It treats the image as a two-dimensional manifold embedded in a hybrid spatial-feature space. Minimization of the image surface area yields the Beltrami flow. The corresponding diffusion operator is anisotropic ...

We construct an example of a smooth map C → C which vanishes to infinite order at the origin, and such that the ratio of the norm of the z̄ derivative to the norm of the z derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.

In this presentation, we describe an implicit Closest Point Method [3] which allows large, stable time steps for high-order PDEs while retaining the principal benefits of the original method. Example computations (including the in-surface heat equation, reaction-diffusion on surfaces, Laplace–Beltrami eigenmodes, and fourth-order interface motion) on a variety of surfaces demonstrate the effect...

We study Beltrami-type equations with two given complex characteristics. Under certain conditions imposed on the coefficients, we prove theorems existence of homeomorphic ACL-solutions this equation. In addition, under some relatively weak conditions, establish corresponding continuous equation that are logarithmic Hölder in a domain.