A note on the Dirichlet problem for the elliptic linear operator in Sobolev spaces with weight
We prove the almost regularity of the degenerate complex Monge-Ampère equation in a special case.
Let L be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain Ω of RN and having Lipschitz coefficients in Ω. It is shown that the Rellich formula with respect to Ω and L extends to all functions in the domain D = {u ∈ H01(Ω); L(u) ∈ L2(Ω)} of L. This answers a question of A. Chaïra and G. Lebeau.
We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces.
In this note, we consider some elliptic systems on a smooth domain of . By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.
We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate...
We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate...