Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDirichlet spectrum and Green function
205
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $sum 1/lambda_{i}^{rm rad}=int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $sum lambda_{i}^{-2}(Omega)$ where $Omega$ is an extrinsic ball of a proper minimal surface of $mathbb{R}^{3}$. In the second part we show that the first eigenvalue of bounded domains is given by iteration of the Green operator and taking the limit, $lambda_{1}(Omega)=lim_{kto infty} Vert G^k(f)Vert_{2}/Vert G^{k+1}(f)Vert_{2}$ for any function $f>0$. In the third part we obtain explicitly the $L^{1}(Omega, mu)$-momentum spectrum of a bounded domain $Omega$ in terms of its Green operator. In particular, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(Omega, mu)$-momentum spectrum, extending the work of Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant balls.
rate research
Read More
Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that ${0,1}$-form of the induced principle is midway between Moser-Trudingers inequality and Nash-Sobolevs inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolevs/Faber-Krahns eigenvalue/Heat-kernel-upper-bound/Log-Sobolevs inequality on the surfaces with finite total Gauss curvature and quadratic area growth.
This work presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near Wood anomaly frequencies. At these frequencies, one or more grazing Rayleigh waves exist, and the lattice sum for the quasi-periodic Green function ceases to exist. We present a modification of this sum by adding two types of terms to it. The first type adds linear combinations of shifted Green functions, ensuring that the spatial singularities introduced by these terms are located below the grating and therefore outside of the physical domain. With suitable coefficient choices these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. Convergence of arbitrarily high order can be obtained by including sufficiently many shifts. The second type of added terms are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on the this new Green function for the problem of scattering by doubly periodic three-dimensional surfaces at and around Wood frequencies. We believe this is the first solver in existence that is applicable to Wood-frequency doubly periodic scattering problems. We demonstrate the proposed approach by means of applications to problems of acoustic scattering by doubly periodic gratings at various frequencies, including frequencies away from, at, and near Wood anomalies.
In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fr{e}chet mean of a set of Dirichlet distributions is uniquely defined in this geometry.
We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.
The variational problem for the functional $F=frac12|phi^*omega|_{L^2}^2$ is considered, where $phi:(M,g)to (N,omega)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $pi:S^3to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $pi$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metric of the Bergers sphere. It is proved that if $phi^*omega$ is coclosed then $phi$ is a critical point of $F$ and minimizes $F$ in its homotopy class. If $M$ is a compact Riemann surface, it is proved that every critical point of $F$ has $phi^*omega$ coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize $F$ in their homotopy class.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
