In this article, we consider the (double) minimization problem $$\min\left\{P(E;\Omega)+\lambda W_p(E,F):~E\subseteq\Omega,~F\subseteq \mathbb{R}^d,~\lvert E\cap F\rvert=0,~ \lvert E\rvert=\lvert F\rvert=1\right\},$$ where $p\geqslant 1$, $\Omega$ is a (possibly unbounded) domain in $\mathbb{R}^d$, $P(E;\Omega)$ denotes relative perimeter of $E$ and $W_p$ $p$-Wasserstein distance. When unbounde...

Abstract In this article, we prove that, generically in the sense of domain variations, any solution to a nonlinear eigenvalue problem is either nondegenerate or Crandall-Rabinowitz transversality condition that satisfied. We then deduce generically, unbounded Rabinowitz continuum solutions simple analytic curve. The global bifurcation diagram resembles classic model case Gel’fand two dimensions.

This work concerns with the following Choquard equation \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+ u = (\int_{\Omega}\frac{u^2(y)}{|x-y|^{N-2}}dy)u &{\rm{in }}\; \Omega , u\in H_0^1(\Omega), \end{cases} \end{equation*} $\end{document} where \Omega\subseteq \mathbb{R}^{N} is an exterior domain smooth boundary. We prove that has at least one positive solution by variational and ...

We consider the following Helmholtz problem which models the propagation of a wave of frequency ω > 0 and velocity of propagation c > 0 in an unbounded homogeneous medium:  ∆u + k2u = 0 in ΩE, u = g on Γ, lim r→∞ √ r ( ∂u ∂r − iku ) = 0, where k := ω/c is the wave number, ΩE := R\ΩI, with ΩI ⊂ R2 being a simply connected bounded domain with regular boundary Γ, and g ∈ H 1 2 (Γ) is a given...

The weak Neumann problem for the Poisson eqution is studied on Lipschitz domain with compact boundary using the direct integral equation method. The domain is bounded or unbounded, the boundary might be disconnected. The problem leads to a uniquely solvable integral equation in H(∂Ω). It is proved that we can get the solution of this equation using the successive approximation method. AMS class...

An air pollution model is generally described by a system of PDEs on unbounded domain. Transformation of the independent variable is used to convert the problem for nonlinear air pollution on finite computational domain. We investigate the new, degenerated parabolic problem in Sobolev spaces with weights for well-posedness and positivity of the solution. Then we construct a fitted finite volume...

A variety of chaotic flows evolving in relatively high-dimensional spaces are considered. It is shown through the use of an optimal choice of basis functions, which are a consequence of the Karhunen-Loeve procedure, that an accurate description can be given in a relatively low-dimensional space. Particular examples of this procedure, which are presented, are the Ginzburg-Landau equation, turbul...

The perfectly matched layers (PMLs), as a boundary termination over an unbounded spatial domain, are widely used in numerical simulations of wave propagation problems. Given a set of discretization parameters, a procedure to select the PML profiles based on minimizing the discrete reflectivity is established for frequency domain simulations. We, by extending the function class and adopting a di...

Imposing either Dirichlet or Neumann boundary conditions on the of a smooth bounded domain Ω, we study perturbation incurred by voltage potential when conductivity is modified in set small measure. We consider (γ n ) n∈ℕ , sequence perturbed matrices differing from γ 0 background matrix measurable well within domain, and assume -γ )γ -1 )→0 L 1 (Ω). Adapting limit measure, show that general rep...

for simultaneously diagonalizable matrices A,B ∈ C . The unbounded drift term is defined by a skew-symmetric matrix S ∈ R. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain D(Ap) of the generator Ap belonging to the OrnsteinUhlenbeck semigroup coincides with the dom...