Let K ⊂ R be a regular convex cone, let e1, . . . , en ∈ ∂K be linearly independent points on the boundary of a compact affine section of the cone, and let x∗ ∈ K be a point in the relative interior of this section. For k = 1, . . . , n, let lk be the line through the points ek and x ∗, let yk be the intersection point of lk with ∂K opposite to ek, and let zk be the intersection point of lk wit...

In this paper, we study the initial and boundary value problem of the Navier-Stokes equations in the half space. We prove the unique existence of weak solution u ∈ L(R+ × (0, T )) with ∇u ∈ L q 2 loc (R+ × (0, T )) for a short time interval when the initial data h ∈ B − 2 q q (R+) and the boundary data g ∈ L(0, T ;B − 1 q q (R)) + L(R;B − 1 2q q (0, T )) with normal component gn ∈ L (0, T ; Ḃ −...

Pramod N. Achar Spring 2000 In this problem set you will prove an analogue of the Fundamental Theorem of Calculus, Stokes’ Theorem, etc. To set up the context of the problem, let us define an operator div◦ on vector fields on R by div◦(P (x, y)i + Q(x, y)j) = ∂Q ∂x − ∂P ∂y . Let C denote a curve in R with endpoints a and b. Let S denote a surface whose boundary is the closed curve BdS. Let V de...

where F : [0, 1]×R → P(R) is a compact valued multivalued map, P(R) is the family of all subsets of R, k ∈ (0, π 2 ), 0 < t1 < t2 < . . . < tm < 1, Ik ∈ C(R,R) (k = 1, 2, . . . , m), ∆x|t=tk = x(t + k )− x(t − k ), x(t + k ) and x(t − k ) represent the right and left limits of x(t) at t = tk respectively, k = 1, 2, . . . , m. In the literature there are few papers dealing with the existence of ...

discretization of boundary integral equations of potential theory in two dimensions. The algorithm is combined with a scheme that adaptively rearranges the parameterization of the boundary in order to minimize the ranks of the off-diagonal blocks in the discretized operator, thus obviating the need for the user to supply a parameterization r of the boundary for which the distance llr(s) -r(t)ll...

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 ...

In this article, we study an inverse boundary value problem for a Dirac-type equation. Let V be a bundle with a base manifold M which has a Dirac structure with a chirality operator F . We assume that we can observe on the lateral boundary ∂M ×R+ the boundary values of solutions of the Dirac-type equation in M × R+. When the coefficients of the Dirac-type equation are time-independent, i.e. we ...

Let Γ be a lattice in G = SL(n,R) and X = G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γ in X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup ...

The purpose of this note is to bring update results on boundary value problems on Lipschitz domains in R or C. We first discuss the Dirichlet problem, the Neumann problem and the d-Neumann problem in a bounded domain in R. These problems are the prototypes of coercive (or elliptic ) boundary value problems when the boundary of the domain is smooth. When the domain is only Lipschitz, solutions t...