We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane ℂ+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm-Liouville problem and its Titchmarsh–Weyl coefficient.

We consider the eigenvalue problem for the one-dimensional (stationary) Dirac operator with some boundary conditions. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, then the potential is actually zero. The analogous statement for the Schrödinger operator is due to Ambarzumian. The proof is based on the fact that the (generalized) moments of a function cannot have alternating signs unless the moments are zero (see §2).

In this paper, we study the asymptotic decay properties in both spatial and temporal variables for a class of weak and strong solutions, by constructing the weak and strong solutions in corresponding weighted spaces. It is shown that, for the strong solution, the rate of temporal decay depends on the rate of spatial decay of the initial data. Such rates of decay are optimal.

This paper outlines the theoretical background of a new approach towards an accurate and well-conditioned perturbative calculation of Dirichlet-Neumann operators (DNOs) on domains that are perturbations of simple geometries. Previous work on the analyticity of DNOs has produced formulae that, as we have found, are very ill-conditioned. We show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae.

We compute critical groups in semilinear elliptic boundary-value problems in which the nonlinear term may fail to have asymptotic limits at zero and at infinity. As applications, we prove several new existence results.

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

We say that a Banach space E is a continuation space for a given parabolic problem if the E-norm of any non-global solution has to become unbounded. We will prove that for large classes of parabolic systems of two equations, the space E = Lr1 × Lr2 can be a continuation space even though the problem is not locally well posed in E. This stands in contrast with classical results for analogous scalar equations.

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manifold.

In the first part, we study several exterior boundary-value problems covering three types of semilinear equations: elliptic, parabolic and hyperbolic. By a unified approach, we show that these problems share a common critical behaviour. In the second part we prove a blow-up result for an inhomogeneous porous medium equation with the critical exponent, which was left open in a previous paper.

In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.

We study the properties of single boundary spike solutions for the following singularly perturbed problem It is known that at a non-degenerate critical point of the mean curvature function H(P), there exists a single boundary spike solution. In this paper, we show that the single boundary spike solution is unique and moreover it has exactly (N − 1) small eigenvalues. We obtain the exact asymptotics of the small eigenvalues in terms of H(P).

If f is an entire function of order ρ, 0 < ρ < 2−11, it is shown that the Nevanlinna deficiency d(0, f′/f) of the logarithmic derivative of f satisfies For small positive ρ, this result strengthens an earlier estimate of Eremenko et al. concerning a conjecture of Fuchs.

We study a problem from nonlinear optics that leads to an integro-differential equation which can be written as the abstract bifurcation problem Au = λLu + ∇φ(u). For such a type of equation, a general theorem on the bifurcation of solutions from the essential spectrum is proved. The hypothesis that φ is non-negative (as is usually assumed) could be omitted and replaced by other conditions that allow us to use the results for the particular example. We also indicate applications to semilinear elliptic equations.

The Brezis–Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

For every 0 < k < min{m,n} and any linear subspace E of real m × n matrices whose non-zero elements have rank greater than k, we show that there is a maximal extension Emax satisfying the same rank condition, and that the dimension of Emax is not less than (m – k)(n – k). We apply this result to the study of quasiconvex functions defined on the complement E⊥ of E in the form F(X) = f(PE⊥(X)), where PE⊥ is the orthgonal projection to E⊥.

We prove existence results for semilinear elliptic boundary-value problems in both the resonance and non-resonance cases. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities, thus allowing the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense. This work generalizes the results of a previous paper, which dealt exclusively with the ordinary differential equation (ODE) case and relied on ODE techniques.

A generalized Yosida approximation of monotone (and non-monotone) operators in Banach space is introduced. It uses a general potential that is not necessarily the square of the norm. It is therefore advisable to use it in cases where some other more convenient potentials are available, such as in Lp-spaces. As an illustration, the case of Nemyckii operators is considered.

This paper presents a sufficient condition for a one-dimensional Dirac operator with a potential tending to infinity at infinity to have no eigenvalues. It also provides a quick proof (and suggests variations) of a related criterion given by Evans and Harris.

In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the given problem, as established by Chanillo and Kiessling.

For a more general class of Liouville-type problems, we show that, rather than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.