A.F.M. ter Elst

Analysis on Lie groups with polynomial growth

I Introduction.- II General Formalism.- II.1 Lie groups and Lie algebras.- II.2 Subelliptic operators.- II.3 Subelliptic kernels.- II.4 Growth properties.- II.5 Real operators.- II.6 Local bounds on kernels.- II.7 Compact groups.- II.8 Transference method.- II.9 Nilpotent groups.- II.10 De Giorgi estimates.- II.11 Almost periodic functions.- II.12 Interpolation.- Notes and Remarks.- III Structure Theory.- III.1 Complementary subspaces.- III.2 The nilshadow algebraic structure.- III.3 Uniqueness of the nilshadow.- III.4 Near-nilpotent ideals.- III.5 Stratified nilshadow.- III.6 Twisted products.- III.7 The nilshadow analytic structure.- Notes and Remarks.- IV Homogenization and Kernel Bounds.- IV.1 Subelliptic operators.- IV.2 Scaling.- IV.3 Homogenization correctors.- IV.4 Homogenized operators.- IV.5 Homogenization convergence.- IV.6 Kernel bounds stratified nilshadow.- IV.7 Kernel bounds general case.- Notes and Remarks.- V Global Derivatives.- V.1 L2-bounds.- V.1.1 Compact derivatives.- V.1.2 Nilpotent derivatives.- V.2 Gaussian bounds.- V.3 Anomalous behaviour.- Notes and Remarks.- VI Asymptotics.- VI. 1 Asymptotics of semigroups.- VI.2 Asymptotics of derivatives.- Notes and Remarks.- Appendices.- A.1 De Giorgi estimates.- A.2 Morrey and Campanato spaces.- A.3 Proof of Theorem II.10.5.- A.4 Rellich lemma.- Notes and Remarks.- References.- Index of Notation.

The Dirichlet-to-Neumann operator via hidden compactness

Abstract We show that to each symmetric elliptic operator of the form A = − ∑ ∂ k a k l ∂ l + c on a bounded Lipschitz domain Ω ⊂ R d one can associate a self-adjoint Dirichlet-to-Neumann operator on L 2 ( ∂ Ω ) , which may be multi-valued if 0 is in the Dirichlet spectrum of A . To overcome the lack of coerciveness in this case, we employ a new version of the Lax–Milgram lemma based on an indirect ellipticity property that we call hidden compactness. We then establish uniform resolvent convergence of a sequence of Dirichlet-to-Neumann operators whenever the underlying coefficients converge uniformly and the second-order limit operator in L 2 ( Ω ) has the unique continuation property. We also consider semigroup convergence.

Dirichlet forms and degenerate elliptic operators

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on ℝd can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behavior of the corresponding evolution semigroup S t can be described in terms of a function (A, B) ↦ d(A; B) ∈ [0, ∞] over pairs of measurable subsets of ℝd. Then

From Forms to Semigroups

\left| {\left( {\varphi A,S_t \varphi B} \right)} \right| \leqslant e^{ - d(A;B)^2 (4t)^{ - 1} } ||\varphi A||2||\varphi B||2

Square roots of perturbed subelliptic operators on Lie groups

for all t > 0 and all ϕ A ↦ L 2(A), ϕ B ∈ L 2(B). Moreover S t L 2(A) ∈ L 2(A) for all t > 0 if and only if d(A;A c) = ∞ where A c denotes the complement of A.

Diffusion determines the manifold

In this paper, we prove L∞-estimates for solutions of divergence operators in the case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.

A GENERALISATION OF THE FORM METHOD FOR ACCRETIVE FORMS AND OPERATORS

We present a review and some new results on form methods for generating holomorphic semigroups on Hilbert spaces.In particular, we explain how the notion of closability can be avoided.As examples we include the Stokes operator, the Black–Scholes equation, degenerate differential equations and the Dirichlet-to-Neumann operator.

On one-parameter Koopman groups

ABSTRACT In the very influential paper [4] Caffarelli and Silvestre studied regularity of (−Δ)s, 0<s<1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea [15] and Galé et al. [7] gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.