A BOUNDARY VALUE PROBLEM IN THE HYPERBOLIC SPACE

We consider a nonlinear problem for the mean curvature equation in the hyperbolic space with a Dirichlet boundary data g. We find solutions in a Sobolev space under appropriate conditions on g.


Introduction
Let M be the open unit ball in R 3 of center 0 and let (1.1) be the hyperbolic metric on M. Let ⊂ R 2 be a bounded domain with smooth boundary ∂ ∈ C 1,1 , and let (u, v) be the variables in R 2 .We consider in this paper the Dirichlet problem for a function X : → M which satisfies the equation of prescribed mean curvature where H : M → R is a given continuous function, and g ∈ W 2,p ( , R 3 ) for 1 < p < ∞, with g ∞ < 1.
In the above equation X u , X v , and X u ∧X v : → T M are the vector fields given by ∂X k ∂v (u,v)   ∂ ∂x k X(u,v)  , where ∂X 1 ∂v (u,v)  , We remark that if X u and X v are linearly independent, then X( ) ⊂ M is an imbedded submanifold and X u ∧ X v (u, v) is the only vector orthogonal to X( ) at X(u, v) that satisfies, for any z where ω is the volume element of (M, , ), namely If ∇ is the Levi-Civita connection associated to , and k ij : M → R are the Christoffel symbols with (g ij ) = (g ij ) −1 , then a simple computation shows that Let E, F, G : → R be the coefficients of the first fundamental form, and the unit normal N : → T M be given by which is orthogonal to the tangent space {X( )} x for any x = X(u, v).Then, if H : → R is the mean curvature of X( ) we obtain (1.11) Thus, (1.11) is the equation of prescribed mean curvature for an imbedded submanifold of M.

A Dirichlet problem for (1.11)
With the notations of the previous section, we consider the Dirichlet problem (1.2).The equation of prescribed mean curvature for a surface in R 3 has been studied for constant H in [3,5], and for H nonconstant in [1,2].Without loss of generality, we may assume that g is harmonic in .Our existence result reads as follows.
Theorem 2.1.Let c 0 and c 1 be some positive constants to be specified.Then (1.2) is solvable for any g ∈ W 2,p ( , R 3 ) harmonic such that In the proof of Theorem 2.1, we ignore the canonical isomorphism ∂/∂x k | X(u,v) → e k (with {e k } the usual basis of R 3 ), and considering X u , X v ∈ R 3 we may write (1.2) as a system ( , R 3 ) is well defined and a strong solution of (1.2) in W 2,p can be regarded as Y = g + X, where X is a fixed point of T .By the usual a priori bounds for the Laplacian and the compactness of the imbedding W 2,p ( , and the continuity of T follows.On the other hand, if (2.6) for some constant c and the result follows.
Proof of Theorem 2.1.With the notation of the previous lemma, by Schauder fixed point theorem, it suffices to see that From the previous computations, we have (2.7) Moreover, by Poincaré's inequality Thus, a sufficient condition for obtaining (2.9) For R small enough we may fix R 1 = g ∞ +c 0 R < 1, and then the theorem is proved if for some R > 0. As last condition is equivalent to our hypothesis, the result holds.

Regularity of the solutions of problem (1.2)
In this section, we state the following regularity result.
Proof.(a) Let X = f ∈ L p .If p ≥ q, let Z be the unique solution in W 2,q of the problem Z = f , Z| ∂ = g.As (X − Z) = 0 and X = Z on ∂ the result follows.
On the other hand, if p < q, we obtain in the same way that X ∈ W 2,p .For 2 ≤ p < q this implies that X ∈ W 1,2q and the result follows.Now we consider the case p < 2, q.Let p 0 = p and define P. Amster et al. 253 where p * n is the critical Sobolev exponent 2p n /(2 − p n ).Then {p n } is bounded, and X ∈ W 1,2p n for every n.If p n < 2, q for every n, then p n is increasing and taking r = lim n→∞ p n , we obtain that r/(2 − r) = r, a contradiction.Hence, p n ≥ q or q > p n ≥ 2 for some n, and the proof is complete.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning and the result follows from the uniqueness in [4, Theorem 9.15].The general case is now immediate, from [4, Theorem 6.19].

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation