where g : R 7→ R strictly monotone increasing and differentiable, Ω open set with compact closure in R , and D convex closed subset of R. Under the assumption that ∇ū ∈ D a.e. in Ω, there is a unique solution u to (1.1) and we can actually give an explicit representation of u is terms of a Lax-type formula. The solution is clearly Lipschitz continuous because ∇u ∈ ∂D a.e. in Ω. The Euler-Lagran...

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in R, n ≥ 3. Its free boundary is of “catenoid” type. This is a higher dimensional analogy of the HauswirthHelein-Pacard solution [18] in R. The existence of such solution is conjectured in [18, Remark 2.4]. This is the first nontrivial smooth solution to the one phase free boundary problem in high...

We study the initial value problem associated to the BenjaminOno equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces Zs,r = Hs(R) ∩L2(|x|2rdx), s ∈ R, s ≥ 1 and s ≥ r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence proper...

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the L 2 ρ (R d ; R 1) ⊗ L 2 ρ (R d ; R d) valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.

where λis are the eigenvalues of the Hessian D 2u. Namely, any global convex solution to (1.1) in R must be a quadratic polynomial. Recall the classical result, any global convex solution in R to the Laplace equation △u = λ1+ · · ·+λn = c or the Monge-Ampère equation log detD2u = log λ1+ · · ·+ log λn = c must be quadratic. Equation (1.1) originates from special Lagrangian geometry [HL]. The (L...

PURPOSE To evaluate the effects of hypertonic saline solution associated to remote ischemic perconditioning in liver ischemia/reperfusion injury in rats. METHODS 25 male rats (Wistar) were distributed into five groups: Sham group (S); Ischemia/Reperfusion group (I/R) with 30 minutes of liver ischemia; Remote ischemic perconditioning group (Per) with three cycles of 10 minutes of I/R performed...

A new dynamic multi-dimensional index structure called RSR-tree is proposed, in which splitting operations of leaf nodes and non-leaf nodes are performed separatively. RSR-tree retains the sequence ordering of index records in a leaf node of RS-tree and thus reduces the addressing time cost of disk access while reading data files. In addition, it integrates the characteristic of reducing the ov...