Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn vanishing near corners of conductive transmission eigenfunctions
65
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in [5,19] as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.
rate research
Read More
We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical solution of such eigenvalue problems.
Consider the Dirichlet-Laplacian in $Omega:= (0,L)times (0,H)$ and choose another open set $omegasubset Omega$. The estimate $0<C_{omega}leq R_{omega}(u):=frac{Vert uVert^{2}_{L^{2}(omega)}}{Vert uVert^{2}_{L^{2}(Omega)}}leq frac{vol(omega)}{vol(omega)}$, for all the eigenfunctions, is well known. This is no longer true for an inhomogeneous elliptic selfadjoint operator $A$. In this work we create a partition among the set of eigenfunctions: $forall omega$, the eigenfunctions satisfy $R_{omega}>C_{omega}>0,exists omega, omega ot=emptyset$, such that $inf R_{omega}(u)=0$,and we wish to characterize these two sets. For two patterns we give a sufficient condition, sometimes necessary. As our operator corresponds to a layered media we can give another representation of its spectrum: i.e. a subset of points of $Rtimes R$ that leads to the suggested partition and others connected results: micro local interpretation, default measures,... Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added.
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for this class of coefficients under an additional mild assumption on the continuity of the coefficients. The analysis is new and based on the $L^p$ regularity theory for the transmission problem established here. It also involves a subtle application of the spectral theory for the Hilbert Schmidt operators. Our work extends largely known results in the literature which are mainly devoted to the isotropic case with $C^infty$-coefficients.
We consider a wide class of families $(F_m)_{minmathbb{N}}$ of Gaussian fields on $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ defined by [F_m:xmapsto frac{1}{sqrt{|Lambda_m|}}sum_{lambdainLambda_m}zeta_lambda e^{2pi ilangle lambda,xrangle}] where the $zeta_lambda$s are independent std. normals and $Lambda_m$ is the set of solutions $lambdainmathbb{Z}^d$ to $p(lambda)=m$ for a fixed elliptic polynomial $p$ with integer coefficients. The case $p(x)=x_1^2+dots+x_d^2$ is a random Laplace eigenfunction whose law is sometimes called the $textit{arithmetic random wave}$, studied in the past by many authors. In contrast, we consider three classes of polynomials $p$: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except multiples of $x_1^2+x_2^2+x_3^2$, and a wide family of polynomials in many variables. For these classes of polynomials, we study the $(d-1)$-dimensional volume $mathcal{V}_m$ of the zero set of $F_m$. We compute the asymptotics, as $mto+infty$ along certain sequences of integers, of the expectation and variance of $mathcal{V}_m$. Moreover, we prove that in the same limit, $frac{mathcal{V}_m-mathbb{E}[mathcal{V}_m]}{sqrt{text{Var}(mathcal{V}_m)}}$ converges to a std. normal. As in previous works, one reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to $p(lambda)=m$. We need to study the number of such solutions for fixed $m$, the number of quadruples of solutions $(lambda,mu, u,iota)$ satisfying $lambda+mu+ u+iota=0$, ($4$-correlations), and the rate of convergence of the counting measure of $Lambda_m$ towards a certain limiting measure on the hypersurface ${p(x)=1}$. To this end, we use prior results on this topic but also prove a new estimate on correlations, of independent interest.
We consider the problem of optimal distribution of a limited amount of conductive material in systems governed by local and non-local scalar diffusion laws. Of particular interest for these problems is the study of the limiting case, which appears when the amount of available material is driven to zero. Such a limiting process is of both theoretical and practical interest and continues to be a subject of active study. In the local case, the limiting optimization problem is convex and has a well understood basis pursuit structure. Still this local problem is quite challenging both analytically and numerically because it is posed in the space of vector-valued Radon measures. With this in mind we focus on identifying the vanishing material limit for the corresponding nonlocal optimal design problem. Similarly to the local case, the resulting nonlocal problem is convex and has the basis pursuit structure in terms of nonlocal antisymmetric two-point fluxes. In stark contrast with the local case, the nonlocal problem admits solutions in Lebesgue spaces with mixed exponents. When the nonlocal interaction horizon is driven to zero, the ``vanishing material limit nonlocal problems provide a one-sided estimate for the corresponding local measure-valued optimal design problem. The surprising fact is that in order to transform the one-sided estimate into a true limiting process it is sufficient to disregard the antisymmetry requirement on the two point fluxes. This result relies on duality and requires generalizing some of the well known nonlocal characterizations of Sobolev spaces to the case of mixed Lebesgue exponents.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
