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...

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number νn is proportional to the product of the angular and the “surface” quantum numbers. Arranging the wave functions by increasing values of the Laplace-Beltrami spectrum, we obtain the...

We consider hypersphere x = x( u, v, w ) in the four dimensional Euclidean space. calculate Gauss map, and curvatures of it. Moreover, we compute second Laplace-Beltrami operator satisfying Δ II A x, where ϵ Mat (4,4).

We prove an uncertainty principle for certain eigenfunction expansions on L2(R+,w(r)dr) and use it to analogues of theorems Chernoff Ingham Laplace-Beltrami operators compact symmetric spaces, special Hermite operator Cn Rn.

We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E∆ S2m−1 ([0, R)) are uniformly bounded as an operator from H(S) to L(S) for all p ∈ (1,∞). We also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on R × R. Suppose that L is a positive definite, self-adjoint operator acting on L(X,μ), where X is a me...

We introduce a new local meshfree method for the approximation of Laplace–Beltrami operator on smooth surface in $${\mathbb {R}}^3$$ . It is direct that uses radial basis functions augmented with multivariate polynomials. A key element this it does not need an explicit expression surface, which can be simply defined by set scattered nodes. Likewise, require expressions normal vectors or curvatu...

The Fourier coefficients of the non-holomorphic Eisenstein cusp forms Laplace-Beltrami operator are here summed. summation is performed after Dirichlet series related to divisor function, properties meromorphic continuability; a new dependence on Euler’s γ constant also found.

Abstract. For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the LaplaceBeltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be e...