Necessary and sufficient conditions for unique determination of plane domains

We say that a domain U ? ?ⁿ is uniquely determined from the relative metric of its Hausdorff boundary (the relative metric is the extension by continuity of the intrinsic metric of the domain to the boundary) if every domain V ? ?ⁿ with the Hausdorff boundary isometric in the relative metric to the Hausdorff boundary of U is isometric to U too (in the Euclidean metrics). In this article we state some necessary and sufficient conditions for a plane domain to be uniquely determined from the relative metric of its Hausdorff boundary.     

Boundary regularity for a family of overdetermined problems for the Helmholtz equation

If a nonconstant solution u of the Helmholtz equation exists on a bounded domain with u satisfying overdetermined boundary conditions (u and its normal derivative both required to be constant on the boundary), then under certain assumptions the boundary of the domain is proved to be real-analytic. Under weaker assumptions, if a real-analytic portion of the boundary has a real-analytic extension, then that extension must also be part of the boundary. Also, an explicit formula for u is given and a condition (which does not involve u) is given for a bounded domain to have such a solution u defined on it. Both of these last results involve acoustic single- and double-layer potentials.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Boundary point lemmas and overdetermined problems

Two elliptic boundary value problems are considered: a problem of mixed type in a cylindrical domain, and a Dirichlet problem in an annular domain. Under some overdetermined conditions on the boundary gradient, symmetry results for domain and solution are proved. The method of proof involves the classical boundary point lemma by Hopf, as well as a suitable adaptation of it that works well at certain corners.     

Regularity of solutions of boundary value problems for parabolic equations of arbitrary order in weighted Hölder spaces

We consider initial-boundary value problems for a uniformly parabolic equation of arbitrary order 2m in a noncylindrical domain whose lateral boundary is nonsmooth with respect to t. We assume that the lower-order coefficients and the right-hand side of the equation, generally speaking, grow to infinity no more rapidly than some power function when approaching the parabolic boundary of the domain, all coefficients of the equation are locally Hölder, and their Hölder constants can grow near that boundary. We construct a smoothness scale of solutions of such problems in weighted Hölder classes of functions whose higher derivatives may grow when approaching the parabolic boundary of the domain.     

Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values

We prove boundary asymptotics to solutions of weighted p-Laplacian equations that take infinite value on the boundary of a bounded domain. Uniqueness of such solutions would then follow as a consequence. Our results extend previously known results by allowing weights that are unbounded in the domain.     

Initial boundary value problem for a nonconservative system in elastodynamics

This article is concerned with the initial boundary value problem for a nonconservative system of hyperbolic equation appearing in elastodynamics in the space time domain x > 0, t > 0. The number of boundary conditions, to be prescribed at the boundary x = 0, depends on the number of characteristics entering the domain. Because our system is nonlinear, the characteristic speeds depends on the unknown and the direction of the characteristics curves are known apriori. As it is well known, the boundary condition has to be understood in a generalised way. One of the standard way is using vanishing viscosity method. We use this method to construct solution for a particular class of initial and boundary data, namely the initial and boundary datas that lie on the level sets of one of the Riemann invariants.     

Sobolev Trace Theorem and the Dirichlet Problem in the Unit Disk

We give a direct and elementary proof for the trace theorem in L _p -based Sobolev spaces, when the domain is the unit disk. We also consider the Dirichlet boundary problem for the Laplace equation, where the boundary value is a function in the Besov space. The Poisson kernel enables us to solve this problem in the unit disk more easily than in a general domain.     

Existence of weak solutions of a nonlinear parabolic equation in a noncylindrical domain

Consider the mixed boundary value problem ?_tu + L[u] = f with a squareintegrable initial value and with zero boundary values in a domain Q. L[u] is a nonlinear elliptic operator in divergence form, defined on a domain with timedependent boundary. Weak solutions in cylindrical domains are used to construct a weak solution in Q by approximating Q by a system of cylinders. It is shown that this solution is continuously dependent on the initial value.     

On non-Newtonian p-fluids. The pseudo-plastic case

In the following we study a class of stationary Navier-Stokes equations with shear dependent viscosity, under the non-slip (Dirichlet) boundary condition. We consider pseudo-plastic fluids. A fluid is said pseudo-plastic, or shear thinning, if in Eq. (1.1) below one has p<2. We are interested in global (i.e., up to the boundary) regularity results, in dimension n=3, for the second order derivatives of the velocity and the first order derivatives of the pressure. We consider a cubic domain Ω and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω and simultaneously with a flat boundary.     

Evaluation of a boundary integral representation for the conformal mapping of the unit disk onto a simply-connected domain

A representation of the conformal mapping g of the interior or exterior of the unit circle onto a simply-connected domain Ω as a boundary integral in terms of?|_?Ω is obtained, where? :=g ^-l. A product integration scheme for the approximation of the boundary integral is described and analysed. An ill-conditioning problem related to the domain geometry is discussed. Numerical examples confirm the conclusions of this discussion and support the analysis of the quadrature scheme.     

Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

On a boundary analog of the Forelli theorem

A boundary analog of the Forelli theorem for real-analytic functions is established, i.e., it is demonstrated that each real-analytic function f defined on the boundary of a bounded strictly convex domain D in the multidimensional complex space with the one-dimensional holomorphic extension property along families of complex lines passing through a boundary point and intersecting D admits a holomorphic extension to D as a function of many complex variables.     

The Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary

We construct the Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to the wave equation in a domain on whose boundary the wave propagation velocity c(x) degenerates as the square root of the distance from the boundary.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

Unbounded domains in ℂ2 with non-compact automorphisms group

Let D be a domain in ?² such that the orbit of an internal point under the group Aut(D) of the holomorphic automorphisms of D accumulates to a smooth boundary point p. It is well known that if p is a strictly pseudoconvex point, such a local information forces D to be biholomorphic to the ball. We prove that if p is only weakly pseudoconvex of finite type, the domain D need not to be biholomorphic to a natural standard model. On the other end our results show that, under some convexity assumptions, D may be realized as a bounded domain with real analytic boundary except at most at one point.     

On the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in ${\mathbb{R}^N}$ of given volume. When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k ≥ 3, we prove that in many cases a minimiser cannot be independent of the value of the constant in the boundary condition, or equivalently of the domain’s volume. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions.     

A domain embedding method for mixed boundary value problems

We propose a domain embedding (fictitious domain) method for elliptic equations subject to mixed boundary conditions, and prove the sharp convergence rate. The theory provides a unified treatment for Dirichlet, Neumann, and Robin boundary conditions. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Calculation and estimation of the Poisson kernel

We provide a simple method for obtaining boundary asymptotics of the Poisson kernel on a domain in RN.     

Determination of a material constant in the impedance boundary condition for electromagnetic fields

We study a recovery problem for an unknown boundary data at the boundary part in static electromagnetism. Our computational area is a bounded domain Ω⊂Rⁿ with a Lipschitz continuous boundary. The problem for determining the coefficient λ is considered. This coefficient represents one of the ferromagnetic material characteristics occupying this domain. The existence and uniqueness of a weak solution are proved and a numerical method for its recovery is supported by numerical experiments.