In the upper half of the unit ball B = {|x | < 1, x1 > 0}, let u and  (a domain in R+ = {x ∈ R n : x1 > 0} ) solve the following overdetermined problem: 1u = χ in B , u = |∇u| = 0 in B \, u = 0 on 5 ∩ B, where B is the unit ball with center at the origin, χ denotes the characteristic function of , 5 = {x1 = 0}, n ≥ 2, and the equation is satisfied in the sense of distributions. We show (am...

In this paper, we are concerned with the third-order quasilinear ordinary differential equation (Φp(u)) = f(t, u, u′, u′′), 0 < t < 1, with the nonlinear boundary conditions u(0) = 0, g(u′(0), u′′(0)) = A, h(u′(1), u′′(1)) = B or u(0) = C, L(u′(0), u′(1)) = 0, R(u′(0), u′(1), u′′(0), u′′(1)) = 0, where A,B, C ∈ R,Φp(u) = |u|p−2u(p > 1), f : [0, 1]× R → R is continuous, g, h, L : R → R, R : R → ...

We study the correlation between total number of critical points random spherical harmonics and with value in any interval I ⊂ <mml:mi math...

We obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite Prandtl number with rigid lower boundary and stress-free upper boundary. Because of the asymmetric boundary conditions the solutions of the Euler-Lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on th...

We give improved upper bounds on the radius of the largest ball of separable states of an m-qubit system around the maximally mixed state. The ratio between the upper bound and the best known lower bound (Hildebrand, quant.ph/0601201) thus shrinks to a constant c = p 34/27 ≈ 1.122, as opposed to a term of order √ m logm for the best upper bound known previously (Aubrun and Szarek, quant.ph/0503...

Let R be a hyperbolic Riemann surface with boundary ∂R and suppose that γ : [0, T ] → R ∪ ∂R is a simple curve with γ(0, T ] ⊂ R and γ(0) ∈ ∂R. By lifting Rt = R \ γ(0, t] to the universal covering space of R (which we assume is the upper half-plane H = {z ∈ C : Im[z] > 0}) via the covering map π : H → R, we can define a family of simply-connected domains Dt = π(Rt) ⊂ H. For each t ∈ [0, T ], s...

We consider the system ⎧ ⎪ ⎨ ⎪ ⎩ −Δ p u = λf (v) in Ω −Δ q v = μg(u) in Ω u = v = 0 on ∂Ω (I) where Δ p u = div(|∇u| p−2 ∇u), Δ q v = div(|∇v| q−2 ∇v), p, q > 1, Ω is the open unit ball in R N , N ≥ 2 and ∂Ω is its boundary. We establish upper and lower estimates for possible positive solutions of system(I).

In this paper we use the method of upper and lower solutions combined with degree theoretic techniques to prove the existence of multiple positive solutions to semipositone superlinear systems of the form −∆u = g1(x, u, v) −∆v = g2(x, u, v) on a smooth, bounded domain Ω ⊂ R with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical,...

Let C be a set of geometric objects in R. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play a central role in the analysis of many geometric algorithms, and the techniques used to attain these...