Imposing Dirichlet Boundary Conditions Research Articles

Fluctuations for ∇ φ interface model on a wall

We consider ∇ φ interface model on a hard wall. The hydrodynamic large-scale space-time limit for this model is discussed with periodic boundary by Funaki et al. (2000, preprint). This paper studies fluctuations of the height variables around the hydrodynamic limit in equilibrium in one dimension imposing Dirichlet boundary conditions. The fluctuation is non-Gaussian when the macroscopic interface is attached to the wall, while it is asymptotically Gaussian when the macroscopic interface stays away from the wall. Our basic method is the penalization. Namely, we substitute in the dynamics the reflection at the wall by strong drift for the interface when it goes down beyond the wall and show the fluctuation result for such massive ∇ φ interface model. Then, this is applied to prove the fluctuation for the ∇ φ interface model on the wall.

The critical parameter for the heat equation with a noise term to blow up in finite time

Consider the stochastic partial differential equation $$u_t=u_{xx}+u^{\gamma W},$$ where $x\in\mathbf{I}\equiv[0,J], W=W(t,x)$ is 2-parameter white noise, and we assume that the initial function $u(0,x)$ is nonnegative and not identically 0. We impose Dirichlet boundary conditions on u in the interval I. We say that u blows up in finite time, with positive probability, if there is a random time $T < \infty$ such that $$ P(\lim_{t \uparrow T} \sup_{x} u(t, x) = \infty) > 0. $$ It was known that if $\gamma<3/2$, then with probability 1, $u$ does not blowup in finite time. It was also known that there is a positive probability of finite time blowup for $\gamma$ sufficiently large. We show that if $\gamma>3/2$, then there is a positive probability that u blows up in finite time.

Energy levels of classical interacting fields in a finite domain in 1 + 1 dimensions

We study the behaviour of bound energy levels for the case of two classical interacting fields and in a finite domain (box) in 1 + 1 dimensions upon which we impose Dirichlet boundary conditions. The total Lagrangian contains a ( /4) 4 self-interaction and an interaction term given by g 2 2 . We calculate its energy eigenfunctions and its corresponding eigenvalues and study their dependence on the size of the box (L ) as well as on the free parameters of the Lagrangian: mass ratio = M 2 /M 2 , and interaction coupling constants and g . We show that for some configurations of the above parameters, there exist critical sizes of the box for which instability points of the field appear.

CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS

The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon's landmark paper [Bull. Amer. Math. Soc.7 (1982) 447] to non-zero vector potentials and more general configuration spaces.

Capacitary Estimates for the Scattering Phase

Let H≔−12Δ acting in L2(Rn), n≥3. Given a compact obstacle K, let HY be the operator obtained from H by imposing Dirichlet boundary conditions on K; here Y is the complement of K. Let θY(λ) be the scattering phase for the scattering system {HY,H;J}, where J denotes restriction to Y. We show that the variation in θY(λ) (modulo integer multiples of π/2) as K varies may be estimated in terms of the capacity of the symmetric difference of the obstacles. Suppose that K has smooth boundary. Let Hβ≔H+β1K (β>0) and θβ(λ) be the scattering phase for the scattering system {Hβ,H}. We show furthermore that θβ(λ) converges to θY(λ) (again modulo integer multiples of π/2) roughly at the rate β−1/2 as β→∞.

Energy Levels of Interacting Fields in a Box

We study the influence of boundary conditions on energy levels of interacting fields in a box and discuss some consequences when we hange the size of the box. In order to do this we calculate the energy levels of bound states of a scalar massive field χ nteracting with another scalar field φ through the Lagrangian \(\mathcal{L}_{\operatorname{int} } \) = \(\frac{3}{2}g\phi ^2 \mathcal{X}^2 \)> in a one-dimensional box on which we impose Dirichlet boundary conditions. We find that the gap between the bound states changes with the size of the box in a nontrivial way. For the case where the masses of the two fields are equal and for large box the energy levels of Dashen-Hasslacher-Neveu (DHN model) are recovered and we have a kind of boson condensate for the ground state. Below a critical box size \(L \sim 2.93\left( {2\sqrt 2 /M} \right)\) the ground-state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below other critical sizes, \(L \sim \left( {6/10} \right)\left( {2\sqrt 2 /M} \right)\) and \(L \sim 1.17\left( {2\sqrt 2 /M} \right)\), of the box, the ground state and firstexcited state merge in the continuum part of the spectrum.

The heat content asymptotics for variable geometries

Let be a time-dependent family of Riemannian metrics on a manifold M with a smooth boundary. Let be the initial temperature of M and let be the specific heat of M. Impose Dirichlet or Neumann boundary conditions and let be the resulting total heat energy content of M. As , one can expand in an asymptotic series in half integer powers of the parameter t. We determine for in terms of geometric quantities; this extends previous results from the autonomous setting where the metric was independent of the parameter t to a dynamic setting where the metric is permitted to be time dependent.

Timelike T-duality, de Sitter space, large N gauge theories and topological field theory

T-Duality on a timelike circle does not interchange IIA and IIB string theories, but takes the IIA theory to a type $IIB^*$ theory and the IIB theory to a type $IIA^*$ theory. The type $II^*$ theories admit E-branes, which are the images of the type II D-branes under timelike T-duality and correspond to imposing Dirichlet boundary conditions in time as well as some of the spatial directions. The effective action describing an E$n$-brane is the $n$-dimensional Euclidean super-Yang-Mills theory obtained by dimensionally reducing 9+1 dimensional super-Yang-Mills on $9-n$ spatial dimensions and one time dimension. The $IIB^*$ theory has a solution which is the product of 5-dimensional de Sitter space and a 5-hyperboloid, and the E4-brane corresponds to anon-singular complete solution which interpolates between this solution and flat space. This leads to a duality between the large $N$ limit of the Euclidean 4-dimensional U(N) super-Yang-Mills theory and the $IIB^*$ string theory in de Sitter space, and both are invariant under the same de Sitter supergroup. This theory can be twisted to obtain a large $N$ topological gauge theory and its topological string theory dual. Flat space-time may be an unstable vacuum for the type $II^*$ theories, but they have supersymmetric cosmological solutions.

Boundary-driven instability

We analyse a reaction-diffusion system and show that complex spatial patterns can be generated by imposing Dirichlet boundary conditions on one or more of the reactant concentrations. This pattern persists even when the homogeneous steady state with Neumann conditions is stable.

On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S2

For any two-dimensional Riemannian manifold ( M, g) we introduce a new functional, h g , on the space of closed simple nonparametrized curves on M. This functional associates to any simple curve Γ the regularize determinant of the Laplace operator on the manifold obtained by cutting M along Γ and imposing Dirichlet boundary conditions. When M is of genus zero we derive a formula for the variation of h g , we prove that the critical points are conformal circles (i.e., the curves which, with respect to the unique metric of constant curvature 1 in the conformal class { e 2α g:α ∈ C ∞( S 2, R )} of g, have constant geodesic curvature), and that the hessian of the functional at a critical point is nondegenerate in directions normal the critical submanifold (Theorem 1.1). We also construct smooth flows on the space of nonparametrized curves retracting the space onto the critical sub-manifold and show that they are gradient-like for our function. These flows deform a given closed embedded curve on S 2 to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3).

Determining the Nonlinearity in a Parabolic Equation from Boundary Measurements

We consider the equation u = ∇. k(u)∇u for a smoothly bounded region Ω ⊂ IRm and seek to determine the nonlinearity k(•) from boundary observations on a finite time interval. We may proceed either by imposing Dirichlet boundary conditions u = g(x) on Σ := (0, T) × ∂Ω and observing the resulting flux ψ: = k(u)∇u • n or, alternatively, by imposing flux boundary conditions k(u)∇u • n = g(χ) and observing the resulting trace, ψ : = u on (part of) Σ. Under suitable assumptions we show uniqueness for the determination of k from the pair [g,ψ]. The principal element in the argument is a stability result, showing convergence to steady state for the nonlinear equation with data constant in t. This uniqueness argument is nonconstructive but some comments are made as to the construction and convergence of approximations, e.g., by regularization.

Condition Estimates

Condition Estimates