Ellipticity on Manifolds With Edges and Boundary

Ellipticity of a manifold with edges and boundary is connected to boundary and edge conditions that complete corresponding operators to Fredholm operators between weighted Sobolev spaces. We study a new parameter-dependent calculus of elliptic operators, where the interior symbols have specific properties on the boundary. We construct elliptic operators with a prescribed number of edge conditions and obtain isomorphisms in the scale of edge Sobolev spaces. Supported by the Chinese-German Cooperation Program ‘Partial Differential Equations’, NSFC of China and DFG of Germany.     

Inverse Problems of Boundary Value Problems for Annular Vibrating Membranes with Piecewise Smooth Positive Functions in the Boundary Conditions

The asymptotic expansions of the trace of the heat kernel for small positive t, where λ_ν are the eigenvalues of the negative Laplacian in Rⁿ (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω₁ and a smooth outer boundary ?Ω₂ where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ _j (j=1,…,m) of ?Ω₁ and on the components of ?Ω₂ are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.     

Nonradial Large Solutions of Sublinear Elliptic Equations

We show that the equation Δu = p(x)f(u) has a positive solution on R ^N , N ≥ 3, satisfying <artwork name="GAPA31011ei1"> <artwork name="GAPA31011ei2"> if and only if <artwork name="GAPA31011ei3"> when ψ(r) = min{p(x): |x| = r}. The nondecreasing continuous function f satisfies f(0) = 0, f (s) > 0 for s > 0, and sup _{s ≥ 1} f(s)/s<∞, and the nonnegative continuous function p is required to be asymptotically radial. This extends previous results which required the function p to be constant or radial.     

Computational Results for Seamount Imaging in a Water Waveguide

This article presents our computational results for imaging a seamount in a waveguide. We assume that there is no knowledge of what kind of boundary condition on the unknown seamount. The incident waves are sent from point sources along a straight line and the corresponding scattered fields are measured from a line above the unknown seamount. Analysis and numerical results are presented.     

Boundedness of Solutions of Dirichlet Problem for a Class of Nonlinear Elliptic Equations with Weights

In this article, under some Hypotheses on weights and on coefficients, using a modification of Moser's method, we establish the boundedness of solutions of Dirichlet Problem for a class of nonlinear elliptic equations.     

A variational approach for a generalized emden-fowler equation

Existence of positive solutions of singular boundary value problems related to Emden-Fowler equation is proved. A general minimization theorem in Sobolev spaces is applied.     

Lie point symmetries and some group invariant solutions of the quasilinear equation involving the infinity Laplacian

Using the classical Lie method we obtain the full Lie point symmetry group of the Aronsson equation in two independent variables. Some group invariant solutions of this equation are found and a conjecture on the Lie point symmetry group of the Aronsson equation in Rⁿ is presented.     

Approximation of infinite-dimensional fleming-viot diffusions

The aim of this article is the approximation of a class of infinite-dimensional diffusions occurring in population genetics using a semigroup approach and a suitable sequence of Bernstein-Schnabl type operators which are associated with assigned sequences of positive real numbers.     

Nonlinear Free Boundary Problems with Singular Source Terms

We prove the existence of solutions to nonlinear free boundary problem with singularities at given points.     

Stationary Solutions of the KdV Equation

In this article, as a first step to study the large time behavior of solutions to the KdV equation on a half-line, we prove the existence of the stationary solutions to the corresponding problem. The aim is to concentrate on understanding the influence of boundary conditions.     

The Generalized Cauchy-Riemann-Fueter Equation and Handedness

A generalization of the Cauchy-Riemann condition in complex analysis is described for complex numbers, quaternions and complex quaternions. The generalization called here generalized Cauchy-Riemann-Fueter analycity encompasses not just the left and right-handed versions of quaternion analysis but also generates other variants for complex quaternions. These multiple variants are shown to satisfy an analogue of Cauchy's Theorem and to have similarities with the generalized Cauchy-Riemann conditions that define monogenic functions on R ^{n + 1}; they are also similar to Fueter-type operators and the Moisil-Theodoresco operator. The multiple variants are shown to have an interpretation that unifies analycity into a single definition. Thus left and right-handedness in quaternions are shown to be two sides of the same concept, and likewise for complex quaternions. This is then shown to have possible physical interpretation for example in understanding the nature of chirality and the 'arrow of time'.     

The Dichotomy Problem for Orbital Measures of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.     

Hemivariational Inequalities with a general growth condition: Existence and Approximation

The article deals with the existence and the approximation of nontrivial solutions to a general hemivariational inequality where the potential corresponding to the nonlinear term admits a superquadratic growth. Finally, convergence results for Galerkin type approximation are established. Since the usual techniques based on coerciveness assumptions fail in this case, one makes use of nonsmooth variational arguments. Our approach allows to cover in a unifying way the superlinear and sublinear situations. Improvements and completions of previous results are obtained.     

Bernoulli jets and the zero mean curvature equation

We consider an elliptic PDE problem related with fluid mechanics. We show that level sets of rescaled solutions satisfy the zero mean curvature equation in a suitable weak viscosity sense. In particular, such level sets cannot be touched from below (above) by a convex (concave) paraboloid in a suitably small neighborhood.     

Uniqueness Theorems for Some Free Boundary Problems in Univalent Functions

Let Φ be a bounded positive continuous function in the complex plane C. We will consider the following problem. We can find a simply connected region Ω containing the origin such that the Riemann mapping function f _Ω mapping the unit disk D onto Ω will satisfy <artwork name="GCOV31013eu1"> We show that there is a unique solution which is starlike with respect to the origin when Φ satisfies <artwork name="GCOV31013eu2"> We also show that if in addtion, ?Φ/?|w|>0 and ?log Φ is subharmonic, then Ω will be a convex region.     

General Wentzell Boundary Conditions and Analytic Semigroups on W 1, p (0, 1)

We are concerned with the analyticity of the (C ₀) semigroups generated by the realizations of the Laplacian Δu:=u″ in the spaces C[0, 1] and W ^{1, p} (0, 1) with the general Wentzell boundary conditions Δu(j)+β ^ju″(j)+γ _ju(j)=0 for j=0,1. Here 1<p<∞ and β _j , γ _j are arbitrary complex numbers for j=0,1.     

Regularity results for the quasi–static Bingham variational inequality in dimensions two and three

We consider the variational inequality describing the stationary flow of a Bingham type fluid in bounded domains. Differentiability properties of weak solutions in suitable energy spaces providing existence theorems are studied. We suppose that the volume forces belong to classes of Morrey type and generalize our previous regularity results concerning slow, steady–state flow of Bingham fluids. Received: 12 February 1996; in final form 16 July 1996     

Extracting the Convex Hull of an Unknown Inclusion in the Multilayered Material

We consider the problem of extracting information about an unknown inclusion embedded in a background layered material that has different constant conductivities across finitely many parallel planes, from the Dirichlet-to-Neumann map. For this purpose we construct the exponentially growing solutions of the governing equation for the background material. The leading coefficient of the equation has discontinuity across finitely many planes that are parallel to each other. Using the property of those solutions, we give an extraction formula of the support function of the inclusion from the Dirichlet-to-Neumann map.     

Principal Eigenvalues for Isaacs Operators with Neumann Boundary Conditions

In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded C ² domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given.      

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.