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Y. Pinchover

Publications and source records attributed to Y. Pinchover.

4 recordsLinked to original sources
arXiv2016

Optimal Hardy Weight for Second-Order Elliptic Operator: An Answer to a Problem of Agmon

For a general subcritical second-order elliptic operator $P$ in a domain $Ω\subset \mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - λW$ is subcritical in $Ω$ for all $λ< 1$, null-critical in $Ω$ for $λ= 1$, and supercritical near any neighborhood of infinity in $Ω$ for any $λ> 1$. Moreover, if $P$ is symmetric and $W>0$, then the spectrum and the essential spectrum of $W^{-1}P$ are equal to $[1,\infty)$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation $Pu=0$, the existence of which depends on the subcriticality of $P$ in $Ω$.

math.AP↗
arXiv2015

On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.

math.FA↗
arXiv2013

Ground state alternative for p-Laplacian with potential term

Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1 0$ satisfying $Q^\prime (v)=0$, such that $Q(u_k)\to 0$, and $u_k\to v$ in $L^p_\mathrm{loc}(Ω$). In the latter case, $v$ is (up to a multiplicative constant) the unique positive supersolution of the equation $Q^\prime (u)=0$ in $Ω$, and one has for $Q$ an inequality of Poincaré type: there exists a positive continuous function $W$ such that for every $ψ\in C_0^\infty(Ω)$ satisfying $\int ψv \mathrm{d}x \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^p \mathrm{d}x\le Q(u)+C|\int u ψ\mathrm{d}x|^p$ for all $u\in C_0^\infty(Ω)$. As a consequence, we prove positivity properties for the quasilinear operator $Q^\prime$ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

math.AP↗