We derive boundary conditions for the vorticity equation with solid wall boundaries. The formulation uses a Dirichlet condition for the normal component of vorticity, and Neumann type conditions for the tangential components. In a Galerkin (integral) formulation the tangential condition is natural, i.e. it is enforced by a right-hand side functional and does not impose a boundary constraint on ...

We treat the two-dimensional projection of the (2 + 1) BTZ black hole (known as Achucarro-Ortiz black hole or as (1+1) dilatonic black hole) as a Casimir-type system. The stress tensor of a massless scalar field satisfying Dirichlet boundary conditions on two one-dimensional “walls” (“Dirichlet walls”) is explicitly calculated in three different vacua. Without employing known regularization tec...

We obtain estimates for convergence rates of the eigenelements (λε, uε) for the Laplace operator in a domain Ω ⊂ R3 periodically perforated along a plane γ = Ω ∩ {x1 = 0}. The boundary conditions are of the Dirichlet type on ∂Ω and of the Robin type, involving a large parameter O(ε−κ), on the boundary of the cavities. The small parameter ε denotes the period while the size of each cavity is O(ε...

In this article we study the exact controllability of a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. The fixed endpoint has a Dirichlet-type boundary condition, while the moving end has a Neumann-type condition. When the speed of the moving endpoint is less than the characteristic speed, the exact controllability of this equation is established by Hi...

We consider fully nonlinear parabolic equations on bounded domains under Dirichlet boundary condition. Assuming that the equation and the domain satisfy certain symmetry conditions, we prove that each bounded positive solution of the Dirichlet problem is asymptotically symmetric. Compared with previous results of this type, we do not assume certain crucial hypotheses, such as uniform (with resp...

An impulsive Cohen-Grossberg neural network with time-varying and S-type distributed delays and reaction-diffusion terms is considered. By using Hardy-Poincaré inequality instead of Hardy-Sobolev inequality or just the nonpositivity of the reaction-diffusion operators, under suitable conditions in terms of M-matrices which involve the reaction-diffusion coefficients and the dimension and size o...

For semilinear Gellerstedt equations with Tricomi, Goursat, or Dirichlet boundary conditions, we prove Pohožaev-type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations. A critical-exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic-hyperbolic or degenerate settings where...

The general static solutions of the scalar field equation for the potential V (φ) = − 1 2 M 2 φ 2 + λ 4 φ 4 are determined for a finite domain in (1 + 1) dimensional space-time. A family of real solutions is described in terms of Jacobi Elliptic Functions. We show that the vacuum-vacuum boundary conditions can be reached by elliptic cn-type solutions in a finite domain, such as of the Kink, for...

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the p...

We deal with eigenvalue problems for the Laplacian varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of and Dirichlet complement. By study an Almgren-type frequency function, we derive upper lower bounds variation sharp estimates case strictly star-shaped region.