arXiv subjects

Derek W. Robinson

Publications and source records attributed to Derek W. Robinson.

At least 19 recordsLinked to original sources

arXiv2021

The weighted Hardy constant

Let $Ω$ be a domain in $R^d$ and $d_Γ$ the Euclidean distance to the boundary $Γ$. We investigate whether the weighted Hardy inequality \[ \|d_Γ^{δ/2-1}φ\|_2\leq a_δ\,\|d_Γ^{δ/2}\,(\nablaφ)\|_2 \] is valid, with $δ\geq 0$ and $a_δ>0$, for all $φ\in C_c^1(Γ_r)$ and all small $r>0$ where $Γ_r=\{x\inΩ: d_Γ(x) 1$ but if $δ\in[0,1\rangle$ then $a_δ(Γ)$ can be strictly larger than $2/|δ-1|$. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators.

math.AP↗

arXiv2020

The weighted Hardy inequality and self-adjointness of symmetric diffusion operators

Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$${\!,}$ $d_Γ$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz functions. We assume that $C\sim c\,d_Γ^{\,δ}$ as $d_Γ\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $δ\geq0$. We also assume that there is an $r>0$ and a $ b_{δ,r}>0$ such that the weighted Hardy inequality \[ \int_{Γ_{\!\!r}} d_Γ^{\,δ}\,|\nabla ψ|^2\geq b_{δ,r}^{\,2}\int_{Γ_{\!\!r}} d_Γ^{\,δ-2}\,| ψ|^2 \] is valid for all $ψ\in C_c^\infty(Γ_{\!\!r})$ where $Γ_{\!\!r}=\{x\inΩ: d_Γ(x)<r\}$. We then prove that the condition $(2-δ)/2<b_δ$ is sufficient for the essential self-adjointness of $H$ on $C_c^\infty(Ω)$ with $b_δ$ the supremum over $r$ of all possible $b_{δ,r}$ in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g.\ fractal, boundaries.

math.FA↗

arXiv2017

Hardy and Rellich inequalities on the complement of convex sets

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(Ω)$ where $Ω= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $Γ=\partialΩ$ denote the boundary of $Ω$ and $d_Γ$ the Euclidean distance to $Γ$. We consider weighting functions $c_Ω=c\circ d_Γ$ with $c(s)=s^δ(1+s)^{δ'-δ}$ and $δ,δ'\geq0$. Then the Hardy inequalities take the form \[ \int_Ωc_Ω\,|\nablaφ|^p\geq b_p\int_Ωc_Ω\,d_Γ^{\;-p}\,|φ|^p \] and the Rellich inequalities are given by \[ \int_Ω|Hφ|^p\geq d_p\int_Ω|c_Ω\,d_Γ^{\,-2}φ|^p \] with $H=-\divv(c_Ω\nabla)$. The constants $b_p, d_p$ depend on the weighting parameter $δ,δ'\geq0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.

math.AP↗

arXiv2017

Hardy inequalities, Rellich inequalities and local Dirichlet forms

First the Hardy and Rellich inequalities are defined for the submarkovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain $Ω\subseteq \Ri^d$. The weighting near the boundary $\partial Ω$ can be different from the weighting at infinity. Finally these results are applied to weighted second-order operators on $\Ri^d\backslash\{0\}$ and to a general class of operators of Grushin type.

math.AP↗

arXiv2016

On extensions of local Dirichlet forms

Let $\ce$ be a Dirichlet form on $L_2(X\,;μ)$ where $(X,μ)$ is locally compact $σ$-compact measure space. Assume $\ce$ is inner regular, i.e.\ regular in restriction to functions of compact support, and local in the sense that $\ce(φ,ψ)=0$ for all $φ, ψ\in D(\ce)$ with $φ\,ψ=0$. We construct two Dirichlet forms $\ce_m$ and $\ce_M$ such that $\ce_m\leq \ce\leq \ce_M$. These forms are potentially the smallest and largest such Dirichlet forms. In particular $\ce_m\supseteq \ce_M$, $(\ce_M)_m=\ce_m$ and $(\ce_m)_M=\ce_M$. We analyze the family of local, inner regular, Dirichlet forms $\cf$ which extend $\ce$ and satisfy $\ce_m\leq \cf\leq \ce_M$. We prove that the latter bounds are valid if and only if $\cf_M=\ce_M$, or $\cf_m=\ce_m$, or $D(\ce_M)$ is an order ideal of $D(\cf)$. Alternatively the $\cf$ are characterized by $D(\ce_M)\cap L_\infty(X)$ being an algebraic ideal of $D(\cf)\cap L_\infty(X)$. As an application we show that if $\ce$ and $\cf$ are strongly local then the Ariyoshi--Hino set-theoretic distance is the same for each of the forms $\ce$, $\ce_M$ and $\cf$. If in addition $\ce_m$ is strongly local then it also defines the same distance. Finally we characterize the uniqueness condition $\ce_M=\ce_m$ by capacity estimates.

math.FA↗

arXiv2015

Uniqueness of diffusion on domains with rough boundaries

Let $Ω$ be a domain in $\mathbf R^d$ and $h(φ)=\sum^d_{k,l=1}(\partial_kφ, c_{kl}\partial_lφ)$ a quadratic form on $L_2(Ω)$ with domain $C_c^\infty(Ω)$ where the $c_{kl}$ are real symmetric $L_\infty(Ω)$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in Ω$. Further assume there are $a, δ>0$ such that $a^{-1}d_Γ^δ\,I\le C\le a\,d_Γ^δ\,I$ for $d_Γ\le 1$ where $d_Γ$ is the Euclidean distance to the boundary $Γ$ of $Ω$. We assume that $Γ$ is Ahlfors $s$-regular and if $s$, the Hausdorff dimension of $Γ$, is larger or equal to $d-1$ we also assume a mild uniformity property for $Ω$ in the neighbourhood of one $z\inΓ$. Then we establish that $h$ is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if $δ\ge 1+(s-(d-1))$. The result applies to forms on Lipschitz domains or on a wide class of domains with $Γ$ a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in $\mathbf R^2$ or the complement of a uniformly disconnected set in $\mathbf R^d$.

math.AP↗

arXiv2014

Gaussian bounds, strong ellipticity and uniqueness criteria

Let $h$ be a quadratic form with domain $W_0^{1,2}(\Ri^d)$ given by \[ h(φ)=\sum^d_{i,j=1}(\partial_iφ,c_{ij}\,\partial_jφ) \] where $c_{ij}=c_{ji}$ are real-valued, locally bounded, measurable functions and $C=(c_{ij})\geq 0 $. If $C$ is strongly elliptic, i.e.\ if there exist $λ, μ>0$ such that $λ\,I\geq C\geq μ\,I>0$, then $h$ is closable, the closure determines a positive self-adjoint operator $H$ on $L_2(\Ri^d)$ which generates a submarkovian semigroup $S$ with a positive distributional kernel~$K$ and the kernel satisfies Gaussian upper and lower bounds. Moreover, $S$ is conservative, i.e.\ $S_t\one=\one$ for all $t>0$. Our aim is to examine converse statements. First we establish that $C$ is strongly elliptic if and only if $h$ is closable, the semigroup $S$ is conservative and $K$ satisfies Gaussian bounds. Secondly, we prove that if the coefficients are such that a Tikhonov growth condition is satisfied then $S$ is conservative. Thus in this case strong ellipticity of $C$ is equivalent to closability of $h$ together with Gaussian bounds on $K$. Finally we consider coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\Ri^d)$. It follows that $h$ is closable and a growth condition of the Täcklind type is sufficient to establish the equivalence of strong ellipticity of $C$ and Gaussian bounds on $K$.

math.AP↗

arXiv2013

The limitations of the Poincar{é} inequality

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_δ\geq H\geq a_2H_δ$ for some $a_1,a_2>0$ where $H_δ$ is a generalized Grušin operator, \[ H_δ=-\nabla_{x_1}\,|x_1|^{(2δ_1,2δ_1')}\,\nabla_{x_1}-|x_1|^{(2δ_2,2δ_2')}\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$, $δ_1,δ_1'\in[0,1\rangle$, $δ_2,δ_2'\geq0$ and $|x_1|^{(2δ,2δ')}=|x_1|^{2δ}$ if $|x_1|\leq 1$ and $|x_1|^{(2δ,2δ')}=|x_1|^{2δ'}$ if $|x_1|\geq 1$. \smallskip We prove that the Poincaré inequality, formulated in terms of the Riemannian geometry corresponding to $H$, is valid if $n\geq 2$, or if $n=1$ and $δ_1\veeδ_1'\in[0,1/2\rangle$ but it fails if $n=1$ and $δ_1\veeδ_1'\in[1/2,1\rangle$. The failure is caused by the leading term. If $δ_1\in[1/2, 1\rangle$ it is an effect of the local degeneracy $|x_1|^{2δ_1}$ but if $δ_1\in[0, 1/2\rangle$ and $δ_1'\in [1/2,1\rangle$ it is an effect of the growth at infinity of $|x_1|^{2δ_1'}$. If $n=1$ and $δ_1\in[1/2, 1\rangle$ then the semigroup $S$ generated by the Friedrichs' extension of $H$ is not ergodic. The subspaces $x_1\geq 0$ and $x_1\leq 0$ are $S$-invariant and the Poincaré inequality is valid on each of these subspaces. If, however, $n=1$, $δ_1\in[0, 1/2\rangle$ and $δ_1'\in [1/2,1\rangle$ then the semigroup $S$ is ergodic but the Poincaré inequality is only valid locally. \smallskip Finally we discuss the implication of these results for the kernel of the semigroup $S$.

math.AP↗

arXiv2013

Analysis of degenerate elliptic operators of Grušin type

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_δ\geq H\geq a_2H_δ$ for some $a_1,a_2>0$ where \[ H_δ=-{\nabla}_{x_1}\cdot(c_{δ_1, δ'_1}(x_1)\,\nabla_{x_1})-c_{δ_2, δ'_2}(x_1)\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$ and $c_{δ_i, δ'_i}$ are positive measurable functions such that $c_{δ_i, δ'_i}(x)$ behaves like $|x|^{δ_i}$ as $x\to0$ and $|x|^{δ_i'}$ as $x\to\infty$ with $δ_1,δ_1'\in[0,1\rangle$ and $δ_2,δ_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x\,;y)\leq a\,(|B(x\,;t^{1/2})|\,|B(y\,;t^{1/2})|)^{-1/2} \] where $|B(x\,;r)|$ denotes the volume of the ball $B(x\,;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.

math.AP↗

arXiv2010

$L_1$-uniqueness of degenerate elliptic operators

Let $Ω$ be an open subset of $\Ri^d$ with $0\in Ω$. Further let $H_Ω=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(Ω)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\barΩ)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0 0$ where $μ(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_Ω$ is $L_1$-unique, i.e.\ it has a unique $L_1$-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e.\ it has a unique $L_2$-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent with the capacity of the boundary of $Ω$, measured with respect to $H_Ω$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets $A$ of the boundary the set and the order of degeneracy of $H_Ω$ at $A$.

math.AP↗

arXiv2009

Markov uniqueness of degenerate elliptic operators

Let $Ω$ be an open subset of $\Ri^d$ and $H_Ω=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(Ω)$ with domain $C_c^\infty(Ω)$ where the coefficients $c_{ij}\in W^{1,\infty}(Ω)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $Ω$. In particular, $H_Ω$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_Ω$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_Ω$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_Ω(\partialΩ)=0$ where $\capp_Ω(\partialΩ)$ is the capacity of the boundary of $Ω$ measured with respect to $H_Ω$. The second main result establishes that Markov uniqueness of $H_Ω$ is equivalent to the semigroup generated by the Friedrichs extension of $H_Ω$ being conservative.

math.AP↗

arXiv2009

Degenerate elliptic operators in one dimension

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $Hφ=-(c φ')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $ν=ν_+\veeν_-$ where $ν_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $ν\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $ν\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative $ c\in W^{1,\infty}_{\rm loc}(\Ri)$ the corresponding operator $H$ has a unique submarkovian extension.

math.AP↗

arXiv2009

Conservation and invariance properties of submarkovian semigroups

Let ${\cal E}$ be a Dirichlet form on $L_2(X)$ and $Ω$ an open subset of $X$. Then one can define Dirichlet forms ${\cal E}_D$, or ${\cal E}_N$, corresponding to ${\cal E}$ but with Dirichlet, or Neumann, boundary conditions imposed on the boundary $\partialΩ$ of $Ω$. If $S$, $S^D$ and $S^N$ are the associated submarkovian semigroups we prove, under general assumptions of regularity and locality, that $S_tϕ= S^D_tϕ$ for all $ϕ\in L_2(Ω)$ and $t>0$ if and only if the capacity ${\mathop{\rm cap}}_Ω(\partialΩ)$ of $\partial Ω$ relative to $Ω$ is zero. Moreover, if $S$ is conservative, i.e. stochastically complete, then ${\mathop{\rm cap}}_Ω(\partialΩ)=0$ if and only if $S^D$ is conservative on $L_2(Ω)$. Under slightly more stringent assumptions we also prove that the vanishing of the relative capacity is equivalent to $S^D_t ϕ= S^N_t ϕ$ for all $ϕ\in L_2(Ω)$ and $t>0$.

math.AP↗

arXiv2009

Flows and invariance for elliptic operators

Let $S$ be the submarkovian semigroup on $L_2({\bf R}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty}$ coefficients $c_{kl}$. Further let $Ω$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $\sum_{l=1}^d c_{kl} \partial_l$ for all $k$.

math.AP↗

arXiv2009

Ellipticity and Ergodicity

Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $Ω$ be an open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\sum^d_{j=1}c_{ij}\partial_j$.

math.AP↗

arXiv2007

Approximately inner derivations

Let $α$ be an approximately inner flow on a $C^*$ algebra $A$ with generator $δ$ and let $δ_n$ denote the bounded generators of the approximating flows $α^{(n)}$. We analyze the structure of the set \cd=\{x\in D(δ): \lim_{n\to\infty}δ_n(x)=δ(x)\} of pointwise convergence of the generators. In particular we examine the relationship of $\cd$ and various cores related to spectral subspaces.

math.OA↗

arXiv2006

Uniform subellipticity

We establish two global subellipticity properties of positive symmetric second-order partial differential operators on $L_2(\Ri^d)$. First, if $m \in \Ni$ then we consider operators $H_0$ with coefficients in $W^{m+1,\infty}(\Ri^d)$ and domain $D(H_0)=W^{\infty,2}(\Ri^d)$ satisfying the subellipticity property \[ c (ϕ, (I+H_0)ϕ)\geq \|Δ^{γ/2} ϕ\|_2^2 \] for some $c>0$ and $γ\in<0,1]$, uniformly for all $ϕ\in W^{\infty,2}(\Ri^d)$, where $Δ$ denotes the usual Laplacian. Then we prove that $D(H^α) \subseteq D(Δ^{αγ})$ for all $α\in [0,2^{-1} (m + 1 + γ^{-1})>$. Hence there is a $c>0$ such that the norm estimate \[ c \|(I+H)^αϕ\|_2\geq \|Δ^{αγ} ϕ\|_2 \] is valid for all $ϕ\in D(H^α)$ where $H$ denotes the self-adjoint closure of $H_0$. In particular, if the coefficients of $H_0$ are in $C_b^\infty(\Ri^d)$ then the conclusion is valid for all $α\geq0$. Secondly, we prove that if \[ H_0=\sum^N_{i=1}X_i^* X_i, \] where the $X_i$ are vector fields on $\Ri^d$ with coefficients in $C_b^\infty(\Ri^d)$ satisfying a uniform version of Hörmander's criterion for hypoellipticity, then $H_0$ satisfies the subellipticity condition for $γ=r^{-1}$ where $r$ is the rank of the set of vector fields. Consequently $D(H^n) \subseteq D(Δ^{n/r})$ for all $n \in \Ni$, where $H$ is the closure of $H_0$.

math.AP↗

arXiv2006

Analysis of degenerate elliptic operators of Grushin type

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are real symmetric and $a_1H_δ\geq H\geq a_2H_δ$ for some $a_1,a_2>0$ where \[ H_δ=-\nabla_{x_1} c_{δ_1, δ'_1}(x_1) \nabla_{x_1}-c_{δ_2, δ'_2}(x_1) \nabla_{x_2}^2 . \] Here $x_1\in{\bf R}^n$, $x_2\in{\bf R}^m$ and $c_{δ_i, δ'_i}$ are positive measurable functions such that $c_{δ_i, δ'_i}(x)$ behaves like $|x|^{δ_i}$ as $x\to0$ and $|x|^{δ_i'}$ as $x\to\infty$ with $δ_1,δ_1'\in[0,1>$ and $δ_2,δ_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x ;y)\leq a (|B(x ;t^{1/2})| |B(y ;t^{1/2})|)^{-1/2} \] where $|B(x ;r)|$ denotes the volume of the ball $B(x ;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.

math.AP↗