Here D ⊂ R is a bounded domain with Lipschitz boundary ΓD, E is the electric field, H is the magnetic field, x̂ = x/|x|, and n is the unit outer normal to ΓD. The applied current J and the initial conditions E0,H0 are assumed to be supported in the circle BR = {x ∈ R : |x| < R} for some R > 0. The electric permittivity ε and magnetic permeability μ are assumed to be positive constants. We remark...

We report on very recent work on the stabilization of the steady-state solutions to Navier-Stokes equations on an open bounded domain R C Rd, d = 2,3, by either interior, or else boundary control. More precisely, as to the interior case, we obtain that the steadystate solutions to Navier-Stokes equations on R C Rd, d = 2,3, with no-slip boundary conditions, are locally exponentially stabilizabl...

Local convective heat flux J(r) in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in a cylindrical cell filled with water. The measured J(r) in the bulk region shows a different scaling behavior with varying Rayleigh numbers compared with that measured in the plume-dominated regions near the sidewall and near the lower conducting plate. The loca...

Using the method of upper and lower solutions, we prove that the singular boundary value problem, −u = f(u) u in (0, 1), u(0) = 0 = u(1) , has a positive solution when 0 < α < 1 and f : R → R is an appropriate nonlinearity that is bounded below; in particular, we allow f to satisfy the semipositone condition f(0) < 0. The main difficulty of this approach is obtaining a positive subsolution, whi...

a r t i c l e i n f o a b s t r a c t The singular problem of non-stationary Darcy flow in a region containing a narrow channel of width O() and high permeability O (1) is shown to be well approximated by a problem with flow concentrated on a weighted Sobolev space over a lower-dimensional interface. The convergence of the solution as → 0 is proved for both the stationary case and the correspon...

In this paper we study the lower semicontinuous envelope with respect to the L-topology of a class of isotropic functionals with linear growth defined on mappings from the n-dimensional ball into RN that are constrained to take values into a smooth submanifold Y of RN . Let B be the unit ball in R and Y a smooth Riemannian manifold of dimension M ≥ 1, isometrically embedded in R for some N ≥ 2....

We consider bifunctions F : C ×C → R where C is an arbitrary subset of a Banach space. We show that under weak assumptions, monotone bifunctions are locally bounded in the interior of their domain. As an immediate corollary, we obtain the corresponding property for monotone operators. Also, we show that in contrast to maximal monotone operators, monotone bifunctions (maximal or not maximal) can...

We study boundary and finite-size effects in the Abelian sandpile model due to Bak, Tang and Wiesenfeld. In the case of half-plane geometry, the probability iPj (r ) of a unit height at the boundary, and at a distance r inside the sample is found for open and closed boundary conditions. The leading asymptotic form of the correlation functions for the unit heights, tPj, (r), in the strip and hal...

We study the existence of solutions anH-system for a revolution surface without boundary for H depending on the radius f . Under suitable conditions we prove that the existence of a solution is equivalent to the solvability of a scalar equation N(a) = L/√2, where N : ⊂ R+ → R is a function depending on H . Moreover, using the method of upper and lower solutions we prove existence results for so...

We prove that the problem of deciding whether a 2or 3-dimensional simplicial complex embeds into R is NP-hard. This stands in contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in R like unknot or 3-sphere recognition which are in NP∩ co-NP (assuming the generalized Riemann hypothesis). Our reduction encodes a satisfiability ins...