arXiv subjects
Publications and source records attributed to Yehuda Pinchover.
We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrödinger operators afterwards.
We study the behaviour of Hardy-weights for a class of variational quasi-linear elliptic operators of $p$-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.
We present some new results concerning perturbation theory for positive solutions of second-order linear elliptic operators, including further study of the equivalence of positive minimal Green functions and the validity of a Liouville comparison principle for nonsymmetric operators.
We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,γ}$ with $γ\in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to be new even for $C^2$-domains.
For a given subcritical discrete Schrödinger operator $H$ on a weighted infinite graph $X$, we construct a Hardy-weight $w$ which is optimal in the following sense. The operator $H - λw$ is subcritical in $X$ for all $λ< 1$, null-critical in $X$ for $λ= 1$, and supercritical near any neighborhood of infinity in $X$ for any $λ> 1$. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs.
We study Schrödinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schrödinger operators on general weighted graphs.
Let $P$ be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold $M$, and let $V$ be a real valued function which belongs to the class of {\em small perturbation potentials} with respect to the heat kernel of $P$ in $M$. We prove that under some further assumptions (satisfying by a large classes of $P$ and $M$) the positive minimal heat kernels of $P-V$ and of $P$ on $M$ are equivalent. Moreover, the parabolic Martin boundary is stable under such perturbations, and the cones of all nonnegative solutions of the corresponding parabolic equations are affine homeomorphic
Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu = 0 in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive solution, then such a solution is called a ground state of P in M, and P is said to be critical in M. We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and P^{\star}. Under some further assumptions, we describe the behaviour at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace-Beltrami operator on a smooth and complete Riemannian manifold M.
Let $H$ be a Schrödinger operator defined on a noncompact Riemannian manifold $Ω$, and let $W\in L^\infty(Ω;\mathbb{R})$. Suppose that the operator $H+W$ is critical in $Ω$, and let $φ$ be the corresponding Agmon ground state. We prove that if $u$ is a generalized eigenfunction of $H$ satisfying $|u|\leq φ$ in $Ω$, then the corresponding eigenvalue is in the spectrum of $H$. The conclusion also holds true if for some $K\Subset Ω$ the operator $H$ admits a positive solution in $\tildeΩ=Ω\setminus K$, and $|u|\leq ψ$ in $\tildeΩ$, where $ψ$ is a positive solution of minimal growth in a neighborhood of infinity in $Ω$. Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.
We improve the classical discrete Hardy inequality \begin{equation*}\label{1} \sum _{n=1}^{\infty }a_{n}^{2}\geq \left({\frac {1}{2}}\right)^{2} \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{2}, \end{equation*} where $\{a_n\}_{n=1}^\infty$ is any sequence of non-negative real numbers.
For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(ϕ_p)$ with their dual spaces, where $1\leq p\leq \infty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(ϕ_p)$ for any $1\leq p\leq \infty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$ on $C_0^\infty$, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if $W$ is a (semi)small perturbation of $L$, then for any $1\leq p\leq \infty$, the associated Green operator is compact on $L^p(ϕ_p)$, and the corresponding spectrum is $p$-independent.
We study qualitative positivity properties of quasilinear equations of the form \[ Q'_{A,p,V}[v] := -\mathrm{div}(|\nabla v|_A^{p-2}A(x)\nabla v) + V(x)|v|^{p-2}v =0 \qquad x\inΩ, \] where $Ω$ is a domain in $\mathbb{R}^n$, $1<p<\infty$, $A=(a_{ij})\in L^\infty_{\rm loc}(Ω;\mathbb{R}^{n\times n})$ is a symmetric and locally uniformly positive definite matrix, $V$ is a real potential in a certain local Morrey space (depending on $p$), and \[ |ξ|_{A}^{2}:=A(x)ξ\cdotξ=\sum_{i,j=1}^n a_{ij}(x)ξ_iξ_j \qquad x\inΩ,~ξ=(ξ_1,\ldots,ξ_n)\in \mathbb{R}^n. \] Our assumptions on the coefficients of the operator for $p\geq 2$ are the minimal (in the Morrey scale) that ensure the validity of the local Harnack inequality and hence the Hölder continuity of the solutions. For some of the results of the paper we need slightly stronger assumptions when $p<2$. We prove an Allegretto-Piepenbrink-type theorem for the operator $Q'_{A,p,V}$, and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case $1<p\leq n,$ we examine the behavior of a positive solution near a nonremovable isolated singularity and characterize the existence of the positive minimal Green function for the operator $Q'_{A,p,V}[u]$ in $Ω$.
Let $Ω$ be an open connected cone in $\mathbb{R}^n$ with vertex at the origin. Assume that the operator $$P_μ:=-Δ-\fracμ{δ_Ω^2(x)}$$ is {\em subcritical} in $Ω$, where $δ_Ω$ is the distance function to the boundary of $Ω$ and $μ\leq 1/4$. We show that under some smoothness assumption on $Ω$, the following improved Hardy-type inequality \begin{equation*} \int_Ω|\nabla φ|^2\,\mathrm{d}x -μ\int_Ω \frac{|φ|^2}{δ_Ω^2}\,\mathrm{d}x \geq λ(μ)\int_Ω \frac{|φ|^2}{|x|^2}\,\mathrm{d}x \qquad \forall φ\in C_0^\infty(Ω), \end{equation*} holds true, and the Hardy-weight $λ(μ)|x|^{-2}$ is optimal in a certain definite sense. The constant $λ(μ)>0$ is given explicitly.
We study positive solutions of half-linear second-order elliptic equations of the form $$Q_{A,V}(u):= -\mathrm{div} (|\nabla u|_{A}^{p-2}A(x)\nabla u)+ V(x)|u|^{p-2}u=0 \quad \mbox{in }Ω,$$ where $1<p<\infty$, $Ω$ is a domain in $\mathbb{R}^{n}$, $n\geq 2$, $V\in L_{\mathrm{loc}}^{\infty}(Ω)$, $A=\big(a_{ij}\big)\in L_{\mathrm{loc}}^{\infty}(Ω,\mathbb{R}^{n^2})$ is a symmetric and locally uniformly positive definite matrix in $Ω$, and $$|ξ|_{A}^{2}:=\left\langle A(x)ξ,ξ\right\rangle=\sum_{i,j=1}^n a_{ij}(x)ξ_iξ_j \qquad x\in Ω, ξ=(ξ_1,\ldots,ξ_n)\in\mathbb{R}^n.$$ We extend criticality theory which has been established for linear operators and for half-linear operators involving the $p$-Laplacian, to the operator $Q_{A,V}$. We prove Liouville-type theorems, and study the behavior of positive solutions of the equation $Q_{A,V}(u)=0$ near an isolated singularity and near infinity in $Ω$, and obtain some perturbations results.
We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-Δ^D_Ω$ in locally twisted three-dimensional tubes $Ω$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, where $E_1$ is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schrödinger operators $-Δ^D_Ω-V$.
Let $\mathcal{Q}(φ):=\int_Ω\big(|\nabla φ|^p+V|φ|^p\big)\dnu$ on $\core$, and assume that $\mathcal{Q}\geq 0$. The aim of the paper is to obtain ''as large as possible" nonnegative (optimal) Hardy-type weight $W$ satisfying $$\mathcal{Q}(φ)\geq \int_Ω W|φ|^p\dnu \quad\forall φ\in\core,$$ on punctured domains $Ω$.
We study the behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u) := -p-Laplacian(u) + V |u|^(p-2) u = 0 in Omega near an isolated singular point zeta, where 1 < p < inf, Omega is a domain in R^d with d > 1, and zeta = 0 or zeta = inf. We obtain removable singularity theorems for positive solutions near zeta. In particular, using a new three-spheres theorems for certain solutions of the above equation near zeta we prove that if V belongs to a certain Kato class near zeta and p>d (respectively, p<d), then any positive solution u of the equation Q'(u)=0 in a punctured neighborhood of zeta=0 (respectively, zeta=inf) is in fact continuous at zeta. Under further assumptions we find the asymptotic behavior of u near zeta.
We discuss a variety of developments in the study of large time behavior of the positive minimal heat kernel of a time independent (not necessarily symmetric) second-order parabolic operator defined on a domain M in $R^d$, or more generally, on a noncompact Riemannian manifold M. Our attention is mainly focused on general results in general settings.