In this note we present a new proof of Sobolevs inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality a
nd to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut-V{e}ron and V{e}rons 1991 classical work.
In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions w
ith a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.
By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic sine-Gordon eq
uation, as well as for the elliptic version of the Ernst equation, can be expressed in terms of the solution of appropriate $2 times 2$-matrix Riemann--Hilbert (RH) problems. These RH problems are defined in terms of certain functions, called spectral functions, which involve the given boundary conditions, but also unknown boundary values. For arbitrary boundary conditions, the determination of these unknown boundary values requires the analysis of a nonlinear Fredholm integral equation. However, there exist particular boundary conditions, called linearizable, for which it is possible to bypass this nonlinear step and to characterize the spectral functions directly in terms of the given boundary conditions. Here, we review the implementation of this effective procedure for the following linearizable boundary value problems: (a) the elliptic sine-Gordon equation in a semi-strip with zero Dirichlet boundary values on the unbounded sides and with constant Dirichlet boundary value on the bounded side; (b) the elliptic Ernst equation with boundary conditions corresponding to a uniformly rotating disk of dust; (c) the elliptic Ernst equation with boundary conditions corresponding to a disk rotating uniformly around a central black hole; (d) the elliptic Ernst equation with vanishing Neumann boundary values on a rotating disk.
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.