As an alternative to the Dirichlet counterterms prescription, I introduce the concept of Kounterterms as the boundary terms with explicit dependence on the extrinsic curvature Kij that regularize the AdS gravity action. Instead of a Dirichlet boundary condition on the metric, a suitable choice of the boundary conditions –compatible with any asymptotically AdS (AAdS) spacetime– ensures a finite ...

We numerically investigate periodic travelling wave solutions for a diffusive predator–prey system with landscape features. The landscape features are modelled through the homogeneous Dirichlet boundary condition which is imposed at the edge of the obstacle domain. To effectively treat the Dirichlet boundary condition, we employ a robust and accurate numerical technique by using a boundary cont...

This article studies the solutions in H1 of a steady transport equation with a divergence-free driving velocity that is W 1,∞, in a two-dimensional bounded polygon. Since the velocity is assumed fully non-homogeneous on the boundary, existence and uniqueness of the solution require a boundary condition on the open part Γ− where the normal component of u is strictly negative. In a previous artic...

We study the condition number of the system matrices that appear in the Boundary Element Method when solving the Stokes equations at a two-dimensional domain. At the boundary of the domain we impose Dirichlet conditions or mixed conditions. We show that for certain critical boundary contours the underlying boundary integral equation is not solvable. As a consequence, the condition number of the...

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: u1t = f1(u2)(∆u1 + a1u1), · · ·, u(n−1)t = fn−1(un)(∆un−1 + an−1un−1), unt = fn(u1)(∆un+anun) with homogenous Dirichlet boundary condition and positive initial condition, where ai (i = 1, 2, ···, n) are positive constants and fi (i = 1, 2, ···, n) satisfy some conditions. The lo...

For a potential function F W R ! RC that attains its global minimum value at two disjoint compact connected submanifolds N in R , we discuss the asymptotics, as ! 0, of minimizers u of the singular perturbed functional E .u/ D R .jruj 2 C 1 2 F.u//dx under suitable Dirichlet boundary data g W @ ! R . In the expansion of E .u / with respect to 1 , we identify the first-order term by the area o...

where ν is the unit normal to ∂Ω and the superscript t indicates transposition (in this case, of the matrix ∇~u = (∂ju)j,α). Relying on the method of layer potentials and suitable Rellich-NečasPayne-Weinberger formulas, the boundary value problems (1.2)-(1.3) with ~ f = 0 and ~g ∈ L(∂Ω), 2− ε < p < 2 + ε, have been treated (in all spacedimensions) by B. Dahlberg, C.Kenig, and G.Verchota (cf. [7...

We prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the Poisson equation with the Dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain. This result represents a weakened form of the acute type condition for the three-d...

In this article we have proposed an accurate finite difference method for approximate numerical solution of second order boundary value problem with Dirichlet boundary conditions. There are numerous numerical methods for solving these boundary value problems. Some these methods are more efficient and accurate than others with some advantages and disadvantages. The results in experiment on model...

where t ≥ 0 denotes time and x ∈ Ω ⊂ R position typical within a d−dimensional bounded domain Ω with “sufficiently smooth” boundary or Ω = R. The states u = (u1, . . . , um), i = 1, . . . , m describe the concentrations or densities of substances/populations ui = ui(t, x). The functions di(t, x) are called diffusion coefficients or diffusivities. On a bounded domain Ω the eqs. (1) have to be co...