We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions in the sense of Baire category, we establish that the corresponding equilibrium problem possesses a unique solution and is well-posed.

We consider the problem of hedging the loss of a given portfolio of derivatives using a set of more liquid derivative instruments. We illustrate why the typical mathematical formulation for this hedging problem is ill-posed. We propose to determine a hedging portfolio by minimizing a proportional cost subject to an upper bound on the hedge risk; this bound is typically slightly larger than the ...

Reconstruction of a signal from its spectral phase or magnitude is in general an ill-posed problem. Various conditions restricting the class of signals under consideration have been shown to be su cient to regularize the problem so that a unique (or essentially unique) signal corresponds to any given spectral magnitude or spectral phase function. This paper shows that a nite discretetime signal...

Theorems on the Fredholm alternative and well-posedness of the Darboux problem ∂2u(t, x) ∂t ∂x = `(u)(t, x) + q(t, x), u(t, x0) = φ(t) for t ∈ [a, b], u(t0, x) = ψ(x) for x ∈ [c, d] are established, where ` : C(D;R) → L(D;R) is a linear bounded operator, q ∈ L(D;R), t0 ∈ [a, b], x0 ∈ [c, d], φ : [a, b]→ R, ψ : [c, d]→ R are absolutely continuous functions, and D = [a, b] × [c, d]. New sufficien...

We analyze a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic PDE system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approxima...

We construct solutions of a free boundary value problem for a hyperbolic equation with Dirichlet boundary data. This problem arises from a model of deformation of granular media.

We prove that the Schrödinger map initial-value problem { ∂ts = s×∆xs on R × [−1, 1]; s(0) = s0 is locally well-posed for small data s0 ∈ H σ0 Q (R ; S), σ0 > (d+ 1)/2, Q ∈ S.

We consider a spherical grain which may be growing by accretion or dissolving in a dilute solution of the same substance where one also has reaction and diiusion. The resulting free boundary problem is shown to be well-posed and additional regularity is obtained.

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in some critical Besov spaces.

We prove an endpoint multilinear estimate for the X spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.