Bulletin of Mathematical Sciences

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper.

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

Perfect C^∗-algebras were introduced by Akeman and Shultz in [Perfect C*-algebras, Mem. Amer. Math. Soc. 55(326) (1985)] and they form a certain subclass of C*-algebras determined by their pure states, and for which the general Stone–Weierstrass conjecture is true. In this paper, we introduce the notion of perfect JC-algebras, and we use the strong relationship between a JC-algebra A and its universal enveloping C^∗-algebra $C^{\ast }(A)$, to establish that if $C^{\ast }(A)$ is perfect and A is of complex type, then A is perfect. It is also shown that every scattered JC-algebra of complex type is perfect, and the same conclusion holds for every JC-algebra of complex type whose primitive spectrum is Hausdorff.

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely ${(-\mathrm{\Delta })}^{\frac{\alpha }{2}}u=V(x)u+g(u,\nabla u)+f$ in ${ℝ}^n$, where $0<\alpha <n$, $g$ is the nonlinearity, $V$ the potential and $f$ is a forcing term. Some examples of nonlinearities dealt with are $u|u{|}^{\rho -1}$, $|\nabla u{|}^{\rho }$ and $\left|u{|}^{{\rho }_1}\right|\nabla u{|}^{{\rho }_2}$, covering large values of $\rho ,{\rho }_1,{\rho }_2$, and particularly variational supercritical powers for $u$ and super-$\alpha$ ones for $|\nabla u|$ (superquadratic if $\alpha =2$). Moreover, we are able to consider some exponential growths, $g$ belonging to certain classes of power series, or $g$ satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

We revisit classical Virtual Element approximations on polygonal and polyhedral decompositions. We also recall the treatment proposed for dealing with decompositions into polygons with curved edges. In the second part of the paper, we introduce a couple of new ideas for the construction of Virtual Element Method (VEM)-approximations on domains with curved boundary, both in two and three dimensions. The new approach looks promising, although sound numerical tests should be made to validate the efficiency of the method.

We compute the Laplacian of the squared norm of the second fundamental form of a surface in Sol₃ and then use this Simons type formula to obtain some gap results for compact constant mean curvature surfaces of this space.