The Picone identity for a class of partial differential equations
The Picone-type identity for the half-linear second order partial differential equation is established and some applications of this identity are suggested.
The point on the simple Molodensky’s problem
Il problema di Molodensky, in approssimazione sferica è detto «semplice» perchè può essere trasformato da problema di derivata obliqua a problema di Dirichlet per l’operatore di Laplace. Tale problema è accuratamente analizzato in questa Nota, con particolare riguardo alla generalizzazione delle condizioni di regolarità soddisfatte dal contorno , sufficienti a garantire l’esistenza di una soluzione fisicamente accettabile.
The point source function of the mixed difference problem in the half-plane
The Poisson equation in homogeneous Sobolev spaces.
The Poisson integral for a ball in spaces of constant curvature
We present explicit expressions of the Poisson kernels for geodesic balls in the higher dimensional spheres and real hyperbolic spaces. As a consequence, the Dirichlet problem for the projective space is explicitly solved. Comparison of different expressions for the same Poisson kernel lead to interesting identities concerning special functions.
The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains.
The positive radial solutions of a class of semilinear elliptic equations.
The prescribed mean curvature equation for a revolution surface with Dirichlet condition.
The principal eigenvalue of the ∞-laplacian with the Neumann boundary condition
We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.
The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition
We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.
The Probabilistic Solution of the Dirichlet Problem for Degenerate Elliptic Equations
We study the Dirichlet problem for degenerate elliptic equations, and show that the probabilistic solution is a unique viscosity solution.
The pseudo--Laplace eigenvalue problem and viscosity solutions as
We consider the pseudo--laplacian, an anisotropic version of the -laplacian operator for . We study relevant properties of its first eigenfunction for finite and the limit problem as .
The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞
We consider the pseudo-p-Laplacian, an anisotropic version of the p-Laplacian operator for . We study relevant properties of its first eigenfunction for finite p and the limit problem as p → ∞.
The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter . The high-frequency (or: semi-classical) parameter is . We let and go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution radiates in the outgoing...
The ranges of nonlinear operators of the polynomial type
The regularity of weak and very weak solutions of the Poisson equation on polygonal domains with mixed boundary conditions (part I)
We examine the regularity of weak and very weak solutions of the Poisson equation on polygonal domains with data in L². We consider mixed Dirichlet, Neumann and Robin boundary conditions. We also describe the singular part of weak and very weak solutions.
The regularity of weak and very weak solutions of the Poisson equation on polygonal domains with mixed boundary conditions (part II)
We examine the regularity of weak and very weak solutions of the Poisson equation on polygonal domains with data in L². We consider mixed Dirichlet, Neumann and Robin boundary conditions. We also describe the singular part of weak and very weak solutions.
The resolution of the bounded curvature conjecture in general relativity
This paper reports on the recent proof of the bounded curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.
The restriction theorem for fully nonlinear subequations
Let be a submanifold of a manifold . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on , restrict to be viscosity subsolutions of the restricted subequation on ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed...
The reverse Hölder inequality for the solution to -harmonic type system.