Degenerate elliptic operators diagonal systems and variational integrals

Using a simple quasiconformal transformation of the independent variables it is shown how some regularity results for weak solutions of quasilinear elliptic systems generalize to several cases where the ellipticity of the principal part degenerates. Similarly it is possible to study the regularity of minima of degenerate variational integrals, as well as elliptic equations and systems in unbounded domains.     

Degenerate Poisson structures in dimension 3

Formal normal forms of degenerate Poisson structures in dimension 3 are described. The main tool of the study is a spectral sequence previously introduced by the author. In particular, this method allows one to obtain a new proof of the linearizability of Poisson structures with semisimple linear part. However, there are nonlinearizable Poisson structures in dimension 3 as well.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 579–592, April, 1998.The author wishes to thank the referee for pointing out reference [3] and for other useful remarks.     

Criteria for ergodicity of Lévy type operators in dimension one

Some sufficient conditions for the recurrence, the positive recurrence and the exponential ergodicity of one-dimensional Lévy type operators are presented. The conditions are classified according to different conditions on the ranges and integrability of the Lévy measure, based on the drift inequalities for the extended generator, and on a comparison with diffusion operators. A number of examples are illustrated, including the fractional Laplacian operator and the Ornstein–Uhlenbeck type operator.     

Degenerate elliptic equations with singular nonlinearities

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of , and compactness holds below a critical dimension N ^#. The nonlinearity f(u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p ≠ 2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blow-up argument and stronger assumptions on the nonlinearity f(u) are required. Authors are partially supported by MIUR, project “Variational methods and nonlinear differential equations”.     

Degenerate elliptic inequalities with critical growth

This article is motivated by the fact that very little is known about variational inequalities of general principal differential operators with critical growth.The concentration compactness principle of P.L. Lions [P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case I, Rev. Mat. Iberoamericana 1 (1) (1985) 145-201; P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case II, Rev. Mat. Iberoamericana 1 (2) (1985) 45-121] is a widely applied technique in the analysis of Palais-Smale sequences. For critical growth problems involving principal differential operators Laplacian or p-Laplacian, much has been accomplished in recent years, whereas very little has been done for problems involving more general main differential operators since a nonlinearity is observed between the corresponding functional I(u) and measure μ introduced in the concentration compactness method. In this paper, we investigate a Leray-Lions type operator and behaviors of its c(P.S.) sequence.     

Broadcasting in one dimension

We consider the broadcasting problem for one-dimensional grid graphs with a given neighborhood template. There are two different models that have been considered-shouting (a node informs all of its neighbors in one step) and whispering (a node informs a single neighbor in one step). Let σ(t) (respectively ω(t)) denote the maximum number of nodes that can be reached in t steps by shouting (respectively whispering) broadcast from a single source.We obtain detailed information about the benefits of shouting over whispering. We prove for the one-dimensional case a conjecture by Stout that ω(t) eventually becomes a polynomial. In particular, we show that there exist constants i and t₀ such that ω(t)=σ(t)−i for all t ≥ t₀. When the broadcast only goes in one direction (i.e., when all elements of the template are positive), we also determine that i=d −1 and t₀≤3d for a neighborhood template with the furthest neighbor at distance d.     

Monodromy of boussinesq elliptic operators

Verdier's program for classifying elliptic operators with a nontrivial centralizer is outlined. Examples of Boussinesq operators are developed.To J.-L. Verdier, in memoriam     

Resolvents of elliptic cone operators

We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent.     

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems. Nicoleta Dines and Bert-Wolfgang Schulze were supported by Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany. Xiaochun Liu was supported by NNSF of China through Grant No. 10501034, and Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany.     

Square roots of elliptic operators

Consider an elliptic sesquilinear form defined on $\text{V}$ × $\text{V}$ by $\text{J[u, v] = ∫}{}_{\mathrm{\Omega }}\text{a}{}_{\mathrm{jk}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ a}{}_{\mathrm{k}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{v}\text{?}\text{+ α}{}_{\mathrm{j}}\text{u}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ au}\text{v}\text{?}\text{dx}$, where $\text{V}$ is a closed subspace of $\text{H}{}^1\text{(Ω)}$ which contains $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$, Ω is a bounded Lipschitz domain in $\text{R}$ⁿ, $\text{a}{}_{\mathrm{jk}}\text{, a}{}_{\mathrm{k}}\text{, α}{}_{\mathrm{j}}\text{, a ? L}{}_{\mathrm{\infty }}\text{(Ω),}\text{and Re}\text{a}{}_{\mathrm{jk}}\text{ζ}{}_{\mathrm{k}}\text{ζ}{}_{\mathrm{j}}\text{? κ > 0}$ for all ζ?$\text{C}$ⁿ with ¦ζ¦ = 1. Let L be the operator with largest domain satisfying J[u, v] = (Lu, v) for all υ∈$\text{V}$. Then L + λI is a maximal accretive operator in $\text{L}{}_2\text{(Ω)}$ for λ a sufficiently large real number. It is proved that $\text{(L + λI)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is a bounded operator from $\text{V}$ to $\text{L}{}_2\text{(Ω)}$ provided mild regularity of the coefficients is assumed. In addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense, then the corresponding square root operators also depend differentiably on t. The latter result is new even when the forms J are hermitian.