arXiv subjects
Publications and source records attributed to A. F. M. ter Elst.
We provide a new short proof for the Birman--Solomyak theorem for Hilbert--Schmidt operators and give an application to a Schrödinger--Poisson system.
We show that the semigroup associated to a second-order elliptic system is positive if and only if the differential equations are essentially decoupled and the coefficients are real-valued. This means the system can be replaced by an equivalent decoupled system (which defines the same semigroup). Instead of matrix coefficients we more generally consider operator-valued coefficients, which leads to interesting additional difficulties with respect to Banach space valued integration.
We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(Ω) \] with possibly complex coefficients. We study three problems: 1) Boundedness on $C^ν$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) Hölder and $L_p$-bounds for the harmonic lifting associated with ${\cal A}$. 3) Poisson bounds for the heat kernel of ${\cal N}$. We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class $C^{1+κ}$ for some $κ> 0$. For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function $G$ of the elliptic operator with Dirichlet boundary conditions.
Nittka gave an efficient criterion on a form defined on $L_2(Ω)$ which implies that the associated semigroup is $L_p$-invariant for some given $p \in (1,\infty)$. We extend this criterion to the Hilbert space valued~$L_2(Ω,H)$. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is $L_p$-contractive for all $p \in [p_-,p_+]$ for some $1 < p_- < 2 < p_+ < \infty$.
In this article we consider the Dirichlet problem on a bounded domain $Ω\subset {\bf R}^d$ with respect to a second-order elliptic differential operator in divergence form. We do not assume a divergence condition as in the pioneering work by Stampacchia, but merely assume that $0$ is not a Dirichlet eigenvalue. The purpose of this article is to define and investigate a solution of the Dirichlet problem, which we call Perron solution, in a setting where no maximum principle is available. We characterise this solution in different ways: by approximating the domain by smooth domains from the interior, by variational properties, by the pointwise boundary behaviour at regular boundary points and by using the approximative trace. We also investigate for which boundary data the Perron solution has finite energy. Finally we show that the Perron solution is obtained as an $H^1_0$-perturbation of a continuous function on $\overline Ω$. This is new even for the Laplacian and solves an open problem.
Let $Ω\subset \mathbb{R}^d$ be a bounded open connected set with Lipschitz boundary. Let $A^N$ and $A^D$ be the Neumann Stokes operator and Dirichlet Stokes operator on $Ω$, respectively. Further let $λ_1^N \leq λ_2^N \leq \ldots$ and $λ_1^D \leq λ_2^D \leq \ldots$ be the eigenvalues of $A^N$ and $A^D$ repeated with multiplicity, respectively. Then \[ λ_{n+1}^N < λ_n^D \] for all $n \in \mathbb{N}$.
We prove a generalized Birman-Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g., a Schrödinger operator) and the associated Birman-Schwinger operator, and additionally offer a careful study of the associated Jordan chains of generalized eigenvectors of both operators. In the course of our analysis we also study algebraic and geometric multiplicities of zeros of strongly analytic operator-valued functions and the associated Jordan chains of generalized eigenvectors. We also relate algebraic multiplicities to the notion of the index of analytic operator-valued functions and derive a general Weinstein-Aronszajn formula for a pair of non-self-adjoint operators.
We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $Ω\subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge δ> 0$ for all $x \in \overlineΩ$, even if the boundary $\partial Ω$ is only Lipschitz continuous. Under such weak regularity assumptions the Hopf-Ole\uınik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map $L_p(Ω)$ into $C(\overlineΩ)$. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.
Consider the elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, c_{kl} \, \partial_l + \sum_{k=1}^d a_k \, \partial_k - \sum_{k=1}^d \partial_k \, b_k + a_0 \] on a bounded connected open set $Ω\subset {\bf R}^d$ with Lipschitz boundary conditions, where $c_{kl} \in L_\infty(Ω,{\bf R})$ and $a_k,b_k,a_0 \in L_\infty(Ω,{\bf C})$, subject to Robin boundary conditions $\partial_νu + β\, {\rm Tr}\, u = 0$, where $β\in L_\infty(\partial Ω, {\bf C})$ is complex valued. Then we show that the kernel of the semigroup generated by $-A$ satisfies Gaussian estimates and Hölder Gaussian estimates. If all coefficients and the function $β$ are real valued, then we prove Gaussian lower bounds. Finally, if $Ω$ is of class $C^{1+κ}$ with $κ> 0$, $c_{kl} = c_{lk}$ is Hölder continuous, $a_k = b_k = 0$ and $a_0$ is real valued, then we show that the kernel of the semigroup associated to the Dirichlet-to-Neumann operator corresponding to $A$ has Hölder Poisson bounds.
Let $Ω\subset {\bf R}^d$ be a bounded open set with Lipschitz boundary $Γ$. It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in $L_2(Ω)$ can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator from $H^{1/2}(Γ)$ into $H^{-1/2}(Γ)$. This result extends the Birman--Schwinger principle in the framework of elliptic operators for the characterization of eigenvalues, eigenfunctions and geometric eigenspaces to the complete set of all generalized eigenfunctions and algebraic eigenspaces.
Let $Ω\subset {\bf R}^d$ be open. We investigate conditions under which an operator $T$ on $L_2(Ω)$ has a continuous kernel $K \in C(\overline Ω\times \overline Ω)$. In the centre of our interest is the condition $T L_2(Ω) \subset C(\overline Ω)$, which one knows for many semigroups generated by elliptic operators. This condition implies that $T^3$ has a kernel in $C(\overline Ω\times \overline Ω)$ if $T$ is self-adjoint and $Ω$ is bounded, and the power $3$ is best possible. We also analyse Mercer's theorem in our context.
Given a complex, elliptic coefficient function we investigate for which values of $p$ the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on $\mathrm{L}^p(Ω)$. Additional properties like analyticity of the semigroup, $\mathrm{H}^\infty$-calculus and maximal regularity are also discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of $p$'s for small imaginary parts of the coefficients. Our results are based on the recent notion of $p$-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.
Let $Ω\subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(Ω) \subset L^r(Ω)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on $Ω$. Let $q > d$ with $\frac{1}{2}-\frac{1}{q} > \frac{1}{r}$. If $p$ is the dual exponent of $q$, then we show that the pre-image of the space $(W^{1,p}(Ω))^*$ under the map $A$ is contained in the space of bounded functions on $Ω$. The considerations are complemented by results on optimal Sobolev regularity for $A$.
We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace spaces and are able to give meaning to the Dirichlet-to-Neumann operator of divergence form operators perturbed by a bounded potential in cases where the boundary of the underlying domain does not allow for a well-defined trace. Moreover, a representation of the Dirichlet-to-Neumann operator as a first-order system of partial differential operators is provided. Using this representation, we address convergence of the Dirichlet-to-Neumann operators in the case that the appropriate reciprocals of the leading coefficients converge in the weak operator topology. We also provide some extensions to the case where the bounded potential is not coercive and consider resolvent convergence.
Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set $Ω\subset R^d$, where $a_{kl}, c_k \in L_\infty(Ω,R)$ and $b_k,c_0 \in L_\infty(Ω,C)$. Suppose that the associated operator on $L_2(Ω)$ with Dirichlet boundary conditions is invertible. Then we show that for all $φ\in C(\partial Ω)$ there exists a unique $u \in C(\overline Ω) \cap H^1_{\rm loc}(Ω)$ such that $u|_{\partial Ω} = φ$ and $A u = 0$. In the case when $Ω$ has a Lipschitz boundary and $φ\in C(\overline Ω) \cap H^{1/2}(\overline Ω)$, then we show that $u$ coincides with the variational solution in $H^1(Ω)$.
We study $L^p$-theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of analytic $C_0$-semigroups under very general conditions on the coefficients, related to the notion of form-boundedness. We determine an interval $J$ in the $L^p$-scale, not necessarily containing $p=2$, in which one obtains a consistent family of quasi-contractive semigroups. This interval is close to optimal, as shown by several examples. In the case of uniform ellipticity we construct a family of semigroups in an extended range of $L^p$-spaces, and we prove $p$-independence of the analyticity sector and of the spectrum of the generators.
Let $Ω$ be a bounded open subset with $C^{1+κ}$-boundary for some $κ> 0$. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator $- \sum \partial_l ( c_{kl} \, \partial_k ) + V$, where the $c_{kl} = c_{lk}$ are Hölder continuous and $V \in L_\infty(Ω)$ are real valued. We prove that the Dirichlet-to-Neumann operator generates a $C_0$-semigroup on the space $C(\partial Ω)$ which is in addition holomorphic with angle $\fracπ{2}$. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence we obtain an optimal holomorphic functional calculus and maximal regularity on $L_p(Γ)$ for all $p \in (1,\infty)$.
Let $Ω\subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $Γ$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(Γ)$ invariant and that the restriction of this semigroup to $C(Γ)$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.