Contents 1. Introduction. 2. Models of PDE 's in the 18th and 19th century. 3. Methods of calculating solutions in the 19th century. 4. Developments of rigorous theories of solvability in the last decades of the 19th century. 5. The period 18901900: the beginning of modern PDE and the work of Poincare. 6. The Hilbert programs. 7. S. Bernstein and the beginning of a priori estimates. 8. Solvabil...

We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman's and Skrypnik's variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik's method, described in [P. Vyridis, {it Bifurcation in...

this study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. our results is depending onthe local minimization method on the nehari manifold and some variational techniques. alsoby using mountain pass lemma, we establish the existence of at least one solution withpositive energy.

We investigate the existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied fractional Laplacian. detect critical separating from nonexistence regimes. For results, we make use particular class weighted Sobolev spaces compensate for singularities which are naturally built in problem.

The purpose of this exercise session is to give an overview of several aspects of scientific computing through the resolution of a few ‘toy problems’. First, we focus on the solving of an elliptic boundary value problem with the Finite Element Method. In a first part, we focus on the implementation of a very basic PDE model, in order to get the reader to be familiar with FreeFem++ software, whi...

The purpose of this exercise session is to give an overview of several aspects of scientific computing. First, we focus on the solving of an elliptic boundary value problem with the Finite Element Method. In a first part, we focus on the implementation of a very basic PDE model, hoping this may get the reader to be familiar with the software FreeFem++, which makes it very easy to handle finite ...

We analyze a second order, linear, elliptic PDE with mixed boundary conditions. This problem arose as a limiting case of a Markovmodulated queueing model for data handling switches in communications networks. We use singular perturbation methods to analyze the problem. In particular we use the ray method to solve the PDE in the e-mail: [email protected] e-mail: [email protected]