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In this paper, we consider an initial boundary value problem for the two-dimensional primitive equations of large scale oceanic dynamics. Assuming that the depth of the ocean is a positive constant, we establish rigorous a priori bounds of the solution to problem. With the aid of these a priori bounds, the continuous dependence of the solution on changes in the boundary terms is obtained.
We consider the general parabolic equation : in with
We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.
A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.
A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.
We determine the convolution operators on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).
We study the semilinear problem with the boundary reaction
where , , is a smooth bounded domain, is a smooth, strictly positive, convex, increasing function which is superlinear at , and is a parameter. It is known that there exists an extremal parameter such that a classical minimal solution exists for , and there is no solution for . Moreover, there is a unique weak solution corresponding to the parameter . In this paper, we continue to study the spectral properties of and show...
This survey of the work of the author with several collaborators presents the way groupoids appear and can be used in index theory. We define the general tools, and apply them to the case of manifolds with corners, ending with a topological index theorem.
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