In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical...

Let $R\left(\lambda \right)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ${\int }_1^T{\left|R\left(t\right)\right|}^{2}dt=c{T}^{\frac{5}{2}}+O_{\delta }\left(T^{\frac{9}{4}+\delta }\right)$, where $c$ is a specific nonzero constant and $\delta$ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R\left(t\right)=O_{\delta }\left(t^{\frac{3}{4}+\delta }\right)$.The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the $2n+1$-dimensional case.

Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition ${\left|\right|u\left|\right|}_{L^s(0,T;L^q\left(\Omega \right))}<\infty$ where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...

Nonlinear Schrödinger equations (NLS)${}_a$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega$ are considered. If $\Omega ={ℝ}^N$ and $a>-{(N-2)}^2/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^2/4$ is excluded because $D\left(P_{a\left(N\right)}^{1/2}\right)$ is not equal to $H^1\left({ℝ}^N\right)$, where $P_{a\left(N\right)}:=-\Delta -{(N-2)}^2/{\left(4\right|x|}^2)$ is nonnegative and selfadjoint in $L^2\left({ℝ}^N\right)$. On the other hand, if $\Omega$ is a smooth and bounded domain with $0\in \Omega$, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....

Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in RN with cylindrical symmetry and with singularities on a whole subspace of RN. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.