In this paper second order elliptic boundary value problems on bounded domains Ω ⊂ Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization pr...

We investigate the eigenvalues of the semi-circulant preconditioned matrix for the finite difference scheme corresponding to the second-order elliptic operator with the variable coefficients given by Lvu := −∆u + a(x, y)ux + b(x, y)uy + d(x, y)u, where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator Lcu is const...

and this index is constant on continuous families of such L. Suppose that D is an elliptic operator of order m on sections of a vector bundle E± over a smooth, compact manifold M . Let H (Γ (M,E±)) denote the Sobolev s-norm completion of the space of sections Γ (M,E), with respect to a chosen metric. Then D can be extended to be a bounded linear operator Ds : H s (Γ (M,E)) → Hs−m (Γ (M,E−)) tha...

We establish Lq bounds on eigenfunctions, and more generally on spectrally localized functions (spectral clusters), associated to a self-adjoint elliptic operator on a compact manifold, under the assumption that the coefficients of the operator are of regularity Cs, where 0 ≤ s ≤ 1. We also produce examples which show that these bounds are best possible for the case q =∞, and for 2 ≤ q ≤ qn.

We consider the question of the essential selfadjointness of a symmetric second order elliptic operator L of general form in the space L2(G) (DL = C ∞ 0 (G)), where G is an arbitrary open set in Rn. The main idea is that using the matrix A(x) of the highest order coefficients of L and the domain G, it is possible to construct a function qA(x) such that the essential selfadjointness of L̄ follows...

We study the geometry of the set of closed extensions of index 0 of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

There is a certain family of conformally invariant first order elliptic systems which include the Dirac operator as its first and simplest member. Their general definition is given and some of their basic properties are described. A special attention is paid to the Rarita-Schwinger operator, the second simplest operator in the row. Its basic properties are described in more details. In the last...

We prove a Liouville-type theorem for bounded stable solutions v ∈ C(R) of elliptic equations of the type (−∆)v = f(v) in R, where s ∈ (0, 1) and f is any nonnegative function. The operator (−∆) stands for the fractional Laplacian, a pseudo-differential operator of symbol |ξ|.

We show that if L is a second-order uniformly elliptic operator in divergence form on R, then C1(1+ |α|) ≤ ‖L‖L1→L1,∞ ≤ C2(1+ |α|). We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

This paper describes a fast corner detection algorithm making use of the Laplacian of Gaussian operator. We propose a general corner model and analyze its behavior in scale space. The study shows that the response of the operator has a stable elliptic extremum which always lies inside the corner. Using a multi-scale representation limited to two scales and the sole laplacian of Gaussian operato...