Krein-like extensions and the lower boundedness problem for elliptic operators

For selfadjoint extensions tilde-A of a symmetric densely defined positive operator A_min, the lower boundedness problem is the question of whether tilde-A is lower bounded {\it if and only if} an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets 1976); this applies to elliptic operators A on bounded domains. For exterior domains, A_gamma ^{-1} is not compact, and whereas the lower bounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from T to tilde-A has only been known when m(T)>-m(A_gamma). We now show it for general T. The operator A_a corresponding to T=aI, generalizing the Krein-von Neumann extension A_0, appears here; its possible lower boundedness for all real a is decisive. We study this Krein-like extension, showing for bounded domains that the discrete eigenvalues satisfy N_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .Comment: 35 pages, revised for misprints and accepted for publication in Journal of Differential Equation

Niches, rather than neutrality, structure a grassland pioneer guild

Pioneer species are fast-growing, short-lived gap exploiters. They are prime candidates for neutral dynamics because they contain ecologically similar species whose low adult density is likely to cause widespread recruitment limitation, which slows competitive dynamics. However, many pioneer guilds appear to be differentiated according to seed size. In this paper, we compare predictions from a neutral model of community structure with three niche-based models in which trade-offs involving seed size form the basis of niche differentiation. We test these predictions using sowing experiments with a guild of seven pioneer species from chalk grassland. We find strong evidence for niche structure based on seed size: specifically large-seeded species produce fewer seeds but have a greater chance of establishing on a per-seed basis. Their advantage in establishment arises because there are more microsites suitable for their germination and early establishment and not directly through competition with other seedlings. In fact, seedling densities of all species were equally suppressed by the addition of competitors' seeds. By the adult stage, despite using very high sowing densities, there were no detectable effects of interspecific competition on any species. The lack of interspecific effects indicates that niche differentiation, rather than neutrality, prevails

Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems

For a Dirac-type operator D with a spectral boundary condition, the associated heat operator trace has an expansion in powers and log-powers of t. Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case. We here investigate the effect of perturbations of D, by use of a pseudodifferential parameter-dependent calculus for boundary problems. It is shown that the first k log-terms are stable under perturbations of D vanishing to order k at the boundary (and the nonlocal power coefficients behind them are only locally perturbed). For perturbations of D from the APS product case by tangential operators commuting with the tangential part A, all the log-coefficients vanish if the dimension is odd.Comment: Published. Abstract added, small typos correcte

Spectral asymptotics for Robin problems with a discontinuous coefficient

The spectral behavior of the difference between the resolvents of two realizations $\tilde A_1$ and $\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $\nu u=b_1\gamma_0u$ and $\nu u=b_2\gamma_0u$, can in the case where all coefficients are $C^\infty$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ of $(\tilde A_1 -\lambda)^{-1}-(\tilde A_2 -\lambda)^{-1}$ satisfy $s_j j^{3/(n-1)}\le C$ for all $j$; this improves a recent result for $A=-\Delta$ by Behrndt et al., that $\sum_js_j ^p(n-1)/3$. A sharper estimate is obtained when $b_1$ and $b_2$ are in $C^\epsilon$ for some $\epsilon >0$, with jumps at a smooth hypersurface, namely that $s_j j^{3/(n-1)}\to c$ for $j\to \infty$, with a constant $c$ defined from the principal symbol of $A$ and $b_2-b_1$. As an auxiliary result we show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators interspersed with piecewise continuous functions.Comment: 20 pages, notation simplified. To appear in J. Spectral Theor

Local and nonlocal boundary conditions for μ\mu-transmission and fractional elliptic pseudodifferential operators

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega$, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega$. As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic $P$ in full scales of Sobolev spaces with a singularity $d^{\mu -k}$, $d(x)=\operatorname{dist}(x,\partial\Omega)$. Examples include fractional Laplacians $(-\Delta)^a$ and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on $R^n\setminus \Omega$ reduce to problems supported on $\bar\Omega$, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces $F^s_{p,q}$ and $B^s_{p,q}$, including H\"older-Zygmund spaces $B^s_{\infty ,\infty}$. This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of $(-\Delta)^au=f\in L_\infty (\Omega)$, $u\in d^aC^a(\bar\Omega)$ when $0<a<1$, $a\ne 1/2$ (in $d^aC^{a-\epsilon}(\bar\Omega)$ when $a=1/2$).Comment: Title slightly changed, 34 page

Remarks on nonlocal trace expansion coefficients

In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a formula for the zero'th coefficient at s=0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-Guillemin noncommutative residue density of an associated operator. The purpose of this note is to provide a proof of that formula relying entirely on resolvent techniques (for the sake of possible generalizations to situations where powers are not an easy tool). - We also give some corrections to transition formulas used in our earlier works.Comment: Minor corrections. To appear in a proceedings volume in honor of K. Wojciechowski, "Analysis and Geometry of Boundary Value Problems", World Scientific, 19 page

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional Laplacian $(-\Delta )^a$ is a particular case. For solutions $u$ of the Dirichlet problem on a bounded smooth subset $\Omega \subset{\Bbb R}^n$, we show an integration-by-parts formula with a boundary integral involving $(d^{-a}u)|_{\partial\Omega }$, where $d(x)=\operatorname{dist}(x,\partial\Omega )$. This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are $x$-dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with $P=(-\Delta +m^2)^a$. The basic step in our analysis is a factorization of $P$, $P\sim P^-P^+$, where we set up a calculus for the generalized pseudodifferential operators $P^\pm$ that come out of the construction.Comment: Final version to appear in J. Differential Equations, 42 pages. References adde

The sectorial projection defined from logarithms

For a classical elliptic pseudodifferential operator P of order m>0 on a closed manifold X, such that the eigenvalues of the principal symbol p_m(x,\xi) have arguments in \,]\theta,\phi [\, and \,]\phi, \theta +2\pi [\, (\theta <\phi <\theta +2\pi), the sectorial projection \Pi_{\theta, \phi}(P) is defined essentially as the integral of the resolvent along {e^{i\phi}R_+}\cup {e^{i\theta}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that \P_{\theta, \phi}(P) is a \psi do of order 0; namely that p_m(x,\xi) cannot in general be modified to allow integration of (p_m(x,\xi)-\lambda)^{-1} along {e^{i\phi}R_+}\cup {e^{i\theta}R_+} simultaneously for all \xi . We show that the structure of \Pi_{\theta, \phi}(P) as a \psi do of order 0 can be deduced from the formula \Pi_{\theta, \phi}(P)= (i/(2\pi))(\log_\theta (P) - \log_\phi (P)) proved in an earlier work (coauthored with Gaarde). In the analysis of \log_\theta (P) one need only modify p_m(x,\xi) in a neighborhood of e^{i\theta}R_+; this is known to be possible from Seeley's 1967 work on complex powers.Comment: Quotations elaborated, 6 pages, to appear in Mathematica Scandinavic