Analysis Seminar

Optimal Hardy-Type Inequality for Nonnegative Second-Order Elliptic Operator: An Answer to a Problem of Shmuel Agmon

Speaker: Yehuda Pinchover, Technion

Location: Warren Weaver Hall 1302

Date: Thursday, December 10, 2015, 11 a.m.

Synopsis:

We give a general answer to the following fundamental problem posed by Shmuel Agmon 30 years ago:

Given a (symmetric) linear elliptic operator \(P\) of second-order in \(\mathbb{R}^n\), find a nonnegative weight function \(W\) which is "as large as possible", such that for some neighborhood of infinity \(G\) the following inequality holds

\((P - W) \geq 0\) in the sense of the associated quadratic form on \(C_0^\infty(G)\).

We construct, for a general subcritical second-order elliptic operator \(P\) on a domain \(D\) in \(\mathbb{R}^n\) (or on a noncompact manifold), a Hardy-type weight \(W\) which is optimal in the following natural sense:

  1. For any \(\lambda \leq 1\), the operator \((P - \lambda W) \geq 0\) on \(C_0^\infty(D)\),
  2. For \(\lambda = 1\), the operator \((P - \lambda W)\) is null-critical in \(D\),
  3. For any \(\lambda > 1\), and any neighborhood of infinity \(G\) of \(D\), the operator \((P - \lambda W)\) is not nonnegative on \(C_0^\infty(G)\).
  4. If \(P\) is symmetric and \(W>0\), then the spectrum and the essential spectrum of the operator \(W^{-1}P\) are equal to \([1,\infty)\).

Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators on a general domain \(D\) or on a noncompact manifold. Moreover, the results can be generalized to certain \(p\)-Laplacian type operators. The constructed weight \(W\) is given by an explicit simple formula involving two positive solutions of the equation \(Pu=0\).

This is a joint work with Baptiste Devyver and Martin Fraas.