On positive solutions of semilinear elliptic problems
In this review article we present an overview on some a priori estimates in , , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two -bounds, , for the solution...
A nonlinear elliptic system involving the p-Laplacian is considered in the whole RN. Existence of nontrivial solutions is obtained by applying critical point theory; also a regularity result is established.
For , precise conditions on the parameters are given under which the particular superposition operator is a bounded map in the Besov space . The proofs rely on linear spline approximation theory.
Analyzing the validity and success of a data assimilation algorithmwhen some state variable observations are not available is an important problem in meteorology and engineering. We present an improved data assimilation algorithm for recovering the exact full reference solution (i.e. the velocity and temperature) of the 3D Planetary Geostrophic model, at an exponential rate in time, by employing coarse spatial mesh observations of the temperature alone. This provides, in the case of this paradigm,...
In this paper we prove the global well-posedness of the two-dimensional Boussinesq system with zero viscosity for rough initial data.
We consider the 3D quantum BBGKY hierarchy which corresponds to the -particle Schrödinger equation. We assume the pair interaction is . For the interaction parameter , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for in [28]. This allows, in the case , for the application of the Klainerman–Machedon uniqueness theorem...
Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be , and in the variable exponent case, L² and -weak.