Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain $H^{2}(\mathbb{R}^{n})$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, $\delta$ and $\delta^{\prime}$-type, assigned either on a $n-1$ dimensional compact boundary $\Gamma=\partial\Omega$ or on a relatively open part $\Sigma\subset\Gamma$. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.Comment: Final revised version, to appear in Journal of Differential Equation

Social Opportunities and Private Convenience of Choices at Farm Level: An Approach to the Links Between Farm Income and Sustainable GDP

This work proposes a method to identify and evaluate the links between the economic and environmental management of a farm, its income, and sustainable GDP. The approach is designed to link micro and macro economic aspects and is based on certain indicators, chosen from among those obtained from analysis of the farm accounts, suitable for representing socially desirable objectives. Three different types of farm accounts are employed. An MADM method of quantitative MCDM analysis was used to make a joint evaluation of various objective indicators in different types of farm management. The work only presents the most interesting result of the research, which was the method itself and does not include the results of a specific case study which was made. This method can be generally applied to connect macro and micro economic aspects and thus might be applicable to different contexts.Method, Farm, Society, Income, Environment, Well-being, Institutional and Behavioral Economics, Q1,

Stable determination of a scattered wave from its far-field pattern: the high frequency asymptotics

We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key tool consists of establishing precise estimates on the behavior of Hankel functions with large argument or order.Comment: 49 page

The equivalent medium for the elastic scattering by many small rigid bodies and applications

We deal with the elastic scattering by a large number $M$ of rigid bodies, $D_m:=\epsilon B_m+z_m$, of arbitrary shapes with $0<\textcolor{black}{\epsilon}<<1$ and with constant Lam\'e coefficients $\lambda$ and $\mu$. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region $\Omega$ of $\mathbb{R}^3$ where their number is $M:=M(\textcolor{black}{\epsilon}):=O(\textcolor{black}{\epsilon}^{-1})$ and the minimum distance between them is $d:=d(\textcolor{black}{\epsilon})\approx \textcolor{black}{\epsilon}^{t}$ with $t$ in some appropriate range, as $\textcolor{black}{\epsilon} \rightarrow 0$, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by $\omega^2\rho I_3-(K+1)\mathbf{C}_0$ where $\rho$ is the density of the background medium (with $I_3$ as the unit matrix) and $(K+1)\mathbf{C}_0$ is the shifting (and possibly variable) coefficient. This shifting coefficient is described by the two coefficients $K$ and $\mathbf{C}_0$ (which have supports in $\overline{\Omega}$) modeling the local distribution of the small bodies and their geometries, respectively. In particular, if the distributed bodies have a uniform spherical shape then the equivalent medium is isotropic while for general shapes it might be anisotropic (i.e. $\mathbf{C}_0$ might be a matrix). In addition, if the background density $\rho$ is variable in $\Omega$ and $\rho =1$ in $\mathbb{R}^3\setminus{\overline{\Omega}}$, then if we remove from $\Omega$ appropriately distributed small bodies then the equivalent medium will be equal to $\omega^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $\Omega$ characterized by $\rho$ is approximately cloaked at the given and fixed frequency $\omega$.Comment: 27pages, 2 figure