On the spectrum of well-defined restrictions and extensions for the Laplace operator

The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $\hat L$ or a Volterra well-defined extension of a minimal operator L ₀ generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator $\hat L$ and of well-defined extensions of the minimal operator L ₀ generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.     

Some extensions of the Cauchy-Davenport theorem

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in $\mathbb{Z}/p\mathbb{Z}$, the cardinality of the sumset $A+B=\{a+b|a\in A,b\in B\}$ is bounded below by $\min (r+s-1,p)$; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers $r,s⩽|G|$, the analogous sharp lower bound, namely the function${\mu }_G(r,s)=\min \{|A+B||A,B\subset G,|A|=r,|B|=s\}.$ Important progress on this topic has been achieved in recent years, leading to the determination of ${\mu }_G$ for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.     

Decay estimates of functions through singular extensions of vector-valued Laplace transforms

Let X be a Banach space and let f∈L^∞(R⁺;X) whose Laplace transform extends analytically to some region containing iR?{0}, possibly having a pole at the origin. In this paper, we give estimates of the decay of certain slight suitable modification of f in terms of the growth of its Laplace transform along the imaginary axis. This technique is applied to obtain decay estimates of smooth orbits of bounded C₀-semigroups whose infinitesimal generators have an arbitrary finite boundary spectrum. These results are close to those given recently by C.J.K. Batty and T. Duyckaerts.     

Some extensions of a hypergraph

It is shown how to construct a vertex-transitive hypergraphX ^* from a suitable collection of isomorphic copies of a given hypergraphX by identifying one or none of the vertices of every two copies in the collection. MoreoverX ^* andX have the same strong chromatic number.If the hypergraphX is edge-coloured, thenX ^* can be so constructed that it is strongly vertex-colour-transitive. This paper also considers the case where a section hypergraph or none of the vertices of every two isomorphic copies is identified in the construction ofX ^*.     

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Some inequalities for Laplace transforms

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Some inequalities for characteristic functions, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, we investigate the counterparts for Laplace transforms of non-negative random variables. Surprisingly, the resulting inequalities hold true on the right half-line. Besides, we show some more inequalities by applying the convex/concave properties of the remainder in Taylor's expansion for the exponential function.     

Solution of the Robin problem for the Laplace equation

For open sets with a piecewise smooth boundary it is shown that we can express a solution of the Robin problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series.     

Some extensions of Bairstow's method

u _xx +u _yy =u _t Bairstow's method for improving an approximate real quadratic factor (x ²?px?q) of a polynomial with real coefficients which leaves a remainderr(x), is to determine δp and δq to satisfy

$$0 = r\left( x \right) + \frac{{\partial r\left( x \right)}}{{\partial P}}\delta P + \frac{{\partial r\left( x \right)}}{{\partial q}}\delta q$$     

Regularization of the Cauchy problem for the Laplace equation

We suggest a method for regularizing the solution of the Cauchy problem for the Laplace equation by introducing the biharmonic operator with a small parameter. We show that if there exists a solution of the original problem, then the difference between the spectral expansions of solutions of the original and regularized equations tends to zero in the space of square integrable functions as the regularization parameter tends to zero. If the original solution belongs to a Sobolev class, then we use results of Il’in’s spectral theory to derive an estimate for the rate of the convergence of the regularized solution to the exact solution.     

The exterior dirichlet problem for the Laplace equation

Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.     

An asymptotic Cauchy problem for the Laplace equation

The Cauchy problem for the Laplace operator $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $$\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $$ with a given majoranth, satisfyingh(+0)=0. Thisasymptotic Cauchy problem only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy dataf, g, and this smoothness is strictly controlled byh. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.     

Solution of Coulomb problem by Laplace transform

The bound state energies and scattering phase shifts for the Coulomb potential are obtained from both the Schrodinger and Dirac equations by taking a Laplace transform. Inversion of transforms is not required, and the nonrelativistic eigenvalue problem is solved without even obtaining the transforms explicitly. The nonrelativistic scattering amplitude appears after solving a first-order differential equation.     

Self-adjoint extensions of the Laplace operator with respect to electric and magnetic boundary conditions

Systems of the type {$\text{A, KJ, 2iK1, I}$} are studied where $\text{J  J1}$ is unitary and $\text{KJK1 }\text{Im}\text{A}$. A complete realization theory as well as state space isomorphism theorem are given. Coupling problems are considered.     

Some extensions of the class of k-convex bodies

We study interrelations between some classes of bodies in Euclidean spaces. We introduce circular projections in normed linear spaces and the classes of bodies related with some families of these projections. Investigation of these bodies more general than k-convex and k-visible bodies allows us to generalize some classical results of geometric tomography and find their new applications.     

Some remarks concerning Riemannian extensions

We study some properties of Riemannian extensions in cotangent bundles with the help of adapted frames.     

Some extensions of weakly mixing flows

Using a technique of R. Ellis we prove the existence of many weakly mixing (w.m.) flows which are distal extensions of a given w.m. flow. Then we indicate two w.m. minimal flows whose product has a minimal non-w.m. subflow. This work was done under the helpful supervision of Professor H. Furstenberg as a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem.