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In his book on convex polytopes [2] A. D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in , n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature problem [20]. In this paper we give a simple variational proof of existence for the A. D. Aleksandrov problem...
In this paper we propose a solution of the Lambertian shape-from-shading
(SFS) problem by designing
a new mathematical framework based on the
notion of viscosity solution. The power of our approach is twofolds:
(1) it defines a notion of weak solutions
(in the viscosity sense) which does not
necessarily require boundary data. Moreover, it allows to characterize the
viscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal.29 (1992) 867–884],...
The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.
We shall prove a weak comparison principle for quasilinear elliptic operators that includes the negative -Laplace operator, where satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.
The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both -Harmonic and -biharmonic operators
is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces and .
In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution of an elliptic equation, that we write , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...
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