Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicity, the squared Laplace o...

Mean value interpolation is a simple, fast, linearly precise method of smoothly interpolating a function given on the boundary of a domain. For planar domains, several properties of the interpolant were established in a recent paper by Dyken and the second author, including: sufficient conditions on the boundary to guarantee interpolation for continuous data; a formula for the normal derivative...

For a function from the Sobolev space H(Ω) definitions of non-unique external and unique internal co-normal derivatives are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain Ω, where they are prescribed, to the domain boundary, where they are not. The notions are then applied to formulation and analysis of direct boun...

Sufficient optimality and sensitivity of a parameterized min-max programming with fixed feasible set are analyzed. Based on Clarke’s subdifferential and Chaney’s second-order directional derivative, sufficient optimality of the parameterized min-max programming is discussed first. Moreover, under a convex assumption on the objective function, a subdifferential computation formula of the margina...

As long ago as 1962 Nitsche [8] conjectured that a harmonic homeomorphism h : A(r,R) onto −→ A(r∗, R∗) between planar annuli exists if and only if R∗ r∗ > 1 2 ` R r + r R ́ . We prove this conjecture when the domain annulus is not too wide; explicitly, when log R r 6 3 2 . For general A(r, R) the conjecture is proved under additional assumption that either h or its normal derivative have vanishi...

This paper develops a formalism for the design of conserving time-integration schemes for Hamiltonian systems with symmetry. The main result is that, through the introduction of a discrete directional derivative, implicit second-order conserving schemes can be constructed for general systems which preserve the Hamiltonian along with a certain class of other first integrals arising from affine s...

640 NOTICES OF THE AMS VOLUME 50, NUMBER 6 H eisenberg groups, in discrete and continuous versions, appear in many parts of mathematics, including Fourier analysis, several complex variables, geometry, and topology. In the present survey we shall not focus too much on any particular aspect, but try to give a kind of sampler. We begin with something which is, in effect, basic calculus. Commutato...

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism h: A(r, R) onto −→A(r∗, R∗) between planar annuli exists if and only if R∗ r∗ 1 2 ( R r + r R ) . We prove this conjecture when the domain annulus is not too wide; explicitly, when R e3/2r. We also treat the general annuli A(r, R), 0 < r < R < ∞, and obtain the sharp Nitsche bound under additional assumption that either h or ...

We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both f...

We design a boundary integral method for time-dependent, threedimensional, doubly periodic water waves and prove that it converges with O(h3) accuracy, without restriction on amplitude. The moving surface is represented by grid points which are transported according to a computed velocity. An integral equation arising from potential theory is solved for the normal velocity. A new method is deve...