Boundary regularity, Pohozaev identities, and nonexistence results

In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main known results of Grubb and of the author and Serra. We also give a simplified proof of one of such results, focusing on the main ideas and on the blow-up techniques that we developed in \cite{RS-K,RS-stable}. After this, we present the Pohozaev identities established in \cite{RS-Poh,RSV,Grubb-Poh} and give a sketch of their proofs, which use strongly the fine boundary regularity results discussed previously. Finally, we show how these Pohozaev identities can be used to deduce nonexistence of solutions or unique continuation properties. The operators $L$ under consideration are integro-differential operator of order $2s$, $s\in(0,1)$, the model case being the fractional Laplacian $L=(-\Delta)^s$.Comment: Survey article. To appear as a chapter in "Recent Developments in the Nonlocal Theory" by De Gruyte

Benjamin Franklin and the birth of a paper money economy

This publication, an essay based on a lecture presented at the Federal Reserve Bank of Philadelphia by Professor Farley Grubb of the University of Delaware, tells readers about Benjamin Franklin’s role in the debate about devising a system of paper money in the colonies and his monetary philosophy.Paper money

Warren County, Kentucky - Court Records (SC 2823)

Finding aid only for Manuscripts Small Collection 2823. Record of the Warren County, Kentucky Court of Quarter Sessions of an action by Jonathan Russell against Jacob Grubb for breach of a contract by Grubb to provide labor. Includes the contract, the plaintiff’s complaint, a warrant and bond

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

Non-commutative residue of projections in Boutet de Monvel's calculus

Using results by Melo, Nest, Schick, and Schrohe on the K-theory of Boutet de Monvel's calculus of boundary value problems, we show that the non-commutative residue introduced by Fedosov, Golse, Leichtnam, and Schrohe vanishes on projections in the calculus. This partially answers a question raised in a recent collaboration with Grubb, namely whether the residue is zero on sectorial projections for boundary value problems: This is confirmed to be true when the sectorial projections is in the calculus.Comment: 10 page

Grubb & Robinson.

R-C of Grubb and Robinson. 23 Jan. SR 113, 54-1, v1, 1p. [3457] Lumber sold to the Prairie band of Pottawatomies in Kansas; 1881-83

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