Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues

We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality due to Muckenhoupt

Lyapunov-type Inequalities for Partial Differential Equations

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the $p-$Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the $p-$Laplacian, and compare them with the usual ones in the literature

Unique continuation property and local asymptotics of solutions to fractional elliptic equations

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.Comment: This is a revision of arXiv:1301.5119v1: some minor changes have been made and Theorem 1.3 has been adde

Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group

Part I: Let (M,g) be a n dimensional smooth, compact, and connected Riemannian manifold without boundary, consider the partial differential equation on M: -Δu=Λu, in which Δ is the Laplace-Beltrami operator. That is, u is an eigenfunction with eigenvalue Λ. We analyze the asymptotic behavior of eigenfunctions as Λ go to ∞ (i.e., limit of high energy states) in terms of the following aspects. (1) Local and global properties of eigenfunctions, including several crucial estimates for further investigation. (2) Write the nodal set of u as N={u=0}, estimate the size of N using Hausdorff measure. Particularly, surrounding the conjecture that the n-1 dimensional Hausdorff measure is comparable to square root of Λ, we discuss separately on lower bounds and upper bounds. (3) BMO (bounded mean oscillation) estimates of eigenfunctions, and local geometric estimates of nodal domains (connected components of nonzero region). (4) A covering lemma which is used in the above estimates, it is of independent interest, and we also propose a conjecture concerning its sharp version. Part II: O the Heisenberg group with homogeneous dimension Q=2n+2, we study the Hardy-Littlewood-Sobolev (HLS) inequality, and particularly its sharp version. Weighted Hardy-Littlewood-Sobolev inequalities with different weights shall also be investigated, and we solve the following problems. (1) Establish the existence results of maximizers. (2) Provide a upper bound of sharp constants