We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary v...

We study the regularity of a parabolic free boundary problem of two-phase type with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the optimal C x ∩ C t -regularity of the solution and that the free boundary is, near the so-called branching points, the union of two graphs that are Lipschitz in time and C in space. In ...

We consider a reconstruction formula for the internal conductivity and uniqueness of conductivity in magnetic resonance electrical impedance tomography (MREIT) which aims to reconstruct the conductivity distribution using internal current distribution. We provide a counter-example of uniqueness for a single measurement of current density with Neumann boundary data and show that at least two mea...

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivity m ≥ 2 thereby extending the results of Christoffel (1867) and Schwarz (1869) for m = 1 and Komatu (1945), m = 2. A formula for f, the conformal map of the exterior of m bounded disks to the exterior of m bounded, disjoint polygons, is derived. The derivation characterizes the global preSchwarzian f...

The Gelfand–Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand–Tsetlin graph has at least three incarnations — as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinitedimensional unitary group, and as the set of doubly infinite totally positive sequences. An ...

Numerical flux formulas for the convection terms in the Euler or Navier-Stokes equations are analyzed with regard to their accuracy in representing steady nonlinear and linear waves (shocks and entropykhear waves, respectively). Numerical results are obtained for a one-dimensional conical Navier-Stokes flow including both a shock and a boundary layer. Analysis and experiments indicate that for ...

This paper concerns with boundary value problems as { L u+a0u = f in Ω, < ∇u, γ > +c0|u|m−1u = g on ∂Ω, where L is an elliptic possibly degenerate second order operator, a0, c0 are positive function, γ is an oblique exterior vector and m 1 . By means of some arguments close to the Dynamics Programming we prove that the viscosity solution admits a representation formula that can be considered as...

We prove a local index formula for cusp-pseudodifferential operators on a manifold with boundary. This is known to be equivalent to an index formula for manifolds with cylindrical ends, and hence we obtain a new proof of the classical Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary, as well as an extension of Melrose’s b-index theorem. Our approach is based on ...

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space we prove that any closed, embedde...

A version of the Atiyah Patodi Singer index theorem is proved for general families of Dirac operators on compact manifolds with boundary The van ishing of the analytic index of the boundary family inK of the base allows us to de ne through an explicit trivialization a smooth family of bound ary conditions of generalized Atiyah Patodi Singer type The calculus of b pseudodi erential operators is ...