We study local boundary regularity for the Stokes system. We show that, unlike in the interior case, non-local effects can lead to a violation of local regularity in the spatial variables near the boundary.

In this paper we solve two biharmonic problems over a square, B = (−1, 1) × (−1, 1). (1) The problem ∇4U = f , for which we determine a particular solution, U , given f , via use of Sinc convolution; and (2) The boundary value problem ∇4V = 0 for which we determine V given V = g and normal derivative Vn = h on ∂B, the boundary of B. The solution to this problem is carried out based on the identity

The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset G of X and x ∈ X \ G, the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 o...

In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is consistent with the classical higher-order Fréchet directional derivative in the sense that both derivatives of the same...

In this paper we consider bilevel programming problems in which the lower level problem is not assumed to have unique optimal solutions for all values of the parameter. Using common approaches to treat such problems, a generally discontinuous function is to be minimized. This leads to the deenition of a new class of functions, so-called generalized PC 1-functions which are composed by a nite nu...

(/? is the birth modulus), that there is no diffusion through the boundary dQ of Q, that is: d A k(a, cc)u(t, a, x) doc = 0 (3) o 8*1 Jc where d/dr] is the normal derivative, and that the initial population is known: w(0, a, x) = u0(a, x). (4) A is the maximum life expectancy of the species. The initial boundary value problem (1)—(4) will be referred to as problem (I), namely (subscripts indica...

Suppose that M is a compact Riemannian manifold with boundary and u is an L-normalized Dirichlet eigenfunction with eigenvalue λ. Let ψ be its normal derivative at the boundary. Scaling considerations lead one to expect that the L norm of ψ will grow as λ1/2 as λ → ∞. We prove an upper bound of the form ‖ψ‖ 2 ≤ Cλ for any Riemannian manifold, and a lower bound cλ ≤ ‖ψ‖ 2 provided that M has no ...

Starting from second-order conditions for C scalar unconstrained optimization problems described in terms of the second-order Dini directional derivative, we pose the problem, whether similar conditions for C vector optimization problems can be derived. We define second-order Dini directional derivatives for vector functions and apply them to formulate such conditions as a Conjecture. The proof...

The non-differentiability and implicit definition of boundary of production possibility set (PPS) in data envelopment analysis (DEA) are two important difficulties for obtaining directional characteristics, including different elasticity measures and marginal rates of substitution. Also, imposing weight restrictions in DEA models have some shortcomings and misunderstandings. In this paper we ut...

The modification of the Clarke generalized subdiNerentia1 due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Glteaux differentiability of any real function can be deduced from the GBteaux differentiability of the norm if the function has a directional derivative which attains a constant related...