arXiv subjects

A. F. M. ter Elst

Publications and source records attributed to A. F. M. ter Elst.

50 records · Page 3Linked to original sources

arXiv2012

Partial Gaussian bounds for degenerate differential operators II

Let $A = - \sum \partial_k \, c_{kl} \, \partial_l$ be a degenerate sectorial differential operator with complex bounded mesaurable coefficients. Let $Ω\subset \mathds{R}^d$ be open and suppose that $A$ is strongly elliptic on $Ω$. Further, let $χ\in C_{\rm b}^\infty(\mathds{R}^d)$ be such that an $\varepsilon$-neighbourhood of $\supp χ$ is contained in $Ω$. Let $ν\in (0,1]$ and suppose that the ${c_{kl}}_{|Ω} \in C^{0,ν}(Ω)$. Then we prove (Hölder) Gaussian kernel bounds for the kernel of the operator $u \mapsto χ\, S_t (χ\, u)$, where $S$ is the semigroup generated by $-A$. Moreover, if $ν= 1$ and the coefficients are real, then we prove Gaussian bounds for the kernel of the operator $u \mapsto χ\, S_t u$ and for the derivatives in the first variable. Finally we show boundedness on $L_p(\mathds{R}^d)$ of various Riesz transforms.

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arXiv2011

Diffusion determines the compact manifold

We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.

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arXiv2011

From forms to semigroups

Form methods give a very efficient tool to solve evolutionary problems on Hilbert space. They were developed by T. Kato [Kat] and, in slightly different language by J.L. Lions. In this expository article we give an introduction based on [AE2]. The main point in our approach is that the notion of closability is not needed anymore. The new setting is particularly efficient for the Dirichlet-to-Neumann operator and degenerate equations. Besides this we give several other examples. This presentation starts by an introduction to holomorphic semigroups. Instead of the contour argument found in the literature, we give a more direct argument based on the Hille--Yosida theorem.

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arXiv2010

Sectorial forms and degenerate differential operators

If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory the assumption that the form is closed or symmetric. Many examples are provided, ranging from complex sectorial differential operators, to Dirichlet-to-Neumann operators and operators with Robin or Wentzell boundary conditions.

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arXiv2010

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set $Ω\subset {\rm R}^d$ whose boundary $Γ$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(Γ)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(Γ)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(Ω)$ which $Ω$ may or may not have.

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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 $t>0$.

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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 $k$.

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arXiv2008

Diffusion determines the manifold

We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.

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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 $H_0$.

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arXiv2006

Contraction semigroups on $L_\infty({\bf R})$

If $X$ is a non-degenerate vector field on ${\bf R}$ and $H=-X^2$ we examine conditions for the closure of $H$ to generate a continuous semigroup on $L_\infty$ which extends to the $L_p$-spaces. We give an example which cannot be extended and an example which extends but for which the real part of the generator on $L_2$ is not lower semibounded.

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arXiv2006

Dirichlet forms and degenerate elliptic operators

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \mapsto d(A ;B)\in[0,\infty]$ over pairs of measurable subsets of $\Ri^d$. Then \[ |(ϕ_A,S_tϕ_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|ϕ_A\|_2\|ϕ_B\|_2 \] for all $t>0$ and all $ϕ_A\in L_2(A)$, $ϕ_B\in L_2(B)$. Moreover $S_tL_2(A)\subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=\infty$ where $A^c$ denotes the complement of $A$.

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arXiv2006

Small time asymptotics of diffusion processes

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.

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arXiv2006

Second-order operators with degenerate coefficients

We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on $\Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p \in [1,\infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Hölder continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in $C^{2-\varepsilon}(\Ri^d)$ with $\varepsilon > 0$.

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arXiv2006

Positivity and strong ellipticity

We consider second-order partial differential operators $H$ in divergence form on $\Ri^d$ with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.

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