Let (M, g) be a smooth, compact Riemannian n-manifold, and h be a Hölder continuous function on M . We prove the existence of multiple changing sign solutions for equations like ∆gu + hu = |u| ∗−2 u, where ∆g is the Laplace–Beltrami operator and the exponent 2∗ = 2n/ (n− 2) is critical from the Sobolev viewpoint.

We find the deformed Heisenberg algebra and the Laplace-Beltrami operator on the extended h-deformed quantum plane and solve the Schrödinger equations explicitly for simple systems on the quantum plane. In the commutative limit the behavior of a quantum particle on the quantum plane becomes that of the quantum particle on the Poincaré half-plane, a surface of constant negative Gaussian curvature.

We study the Dirichlet problem as with continuous boundary data in arbitrary simply connected bounded domains D of complex plane where f satisfies degenerate Beltrami equation a. e. D. give terms BMO and FMO criteria well a number other integral on existence representation regular discrete open solutions to stated above problem.

First, we prove that the BMO condition by John–Nirenberg leads in natural way to asymptotic homogeneity at origin of regular homeomorphic solutions degenerate Beltrami equations. Then, on this basis establish a series criteria for existence equations whole complex plane with infinity. These results can be applied fluid mechanics strongly anisotropic and inhomogeneous media because equation is f...

We report on a new general class of solutions of the Beltrami equation, with special characteristics. We also provide examples of solutions that also satisfy Maxwell equations. A subset of these solutions can be isolated which corresponds to " gauge " fields. A special projective geometry of vacuum fields is also revealed and discussed. 1. Introduction The notion of a force-free field comes dir...

We investigate the Hilbert boundary-value problem for Beltrami equations $$ \overline{\partial}f = ??f with singularities in generalized quasidisks D whose Jordan boundary ?D consists of a countable collection open quasiconformal arcs and, maybe, points. Such quasicircles can be nowhere even locally rectifiable but include, instance, all piecewise smooth curves, as well Lipschitz curves. Genera...

In this paper, we investigate optimal control problems for Allen–Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the Laplace– Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribe...

Chebyshev polynomials of the first and the second kind in n variables z. , Zt , ... , z„ are introduced. The variables z, , z-,..... z„ are the characters of the representations of SL(n + 1, C) corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami op...

Abstract We study the removable singularities for solutions to the Beltrami equation ∂f = μ∂f , where μ is a bounded function, ‖μ‖∞ ≤ K−1 K+1 < 1, and such that μ ∈ W 1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl’s lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly s...

In this paper we discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. A particular example are those of the form f(z)χΩ(z), where Ω is a bounded domain with boundary of class C1+ε and f a function in Lip(ε,Ω) satisfying ‖f‖∞ < 1. An important point is that there is no restriction whatsoever on the Lip(ε,Ω) norm of f besides the requirement on ...