A coercive bilinear form for Maxwell's equations

A coercive bilinear form for Maxwell's equations

When one wants to treat the time-harmonic Maxwell equations with variational methods, one has to face the problem that the natural bilinear form is not coercive on the whole Sobolev space H. On can, however, make it coercive by adding a certain bilinear form on the boundary of the domain. This addition causes a change in the natural boundary conditions. The additional bilinear form (see (2.7), ...

Coercive combined field integral equations

Bilinear Stochastic Elliptic Equations

We study stochastic elliptic PDEs driven by multiplicative Gaussian white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an e ective tool to study such equations and present the corresponding solvability results.

Generalized Bilinear Differential Equations

We introduce a kind of bilinear differential equations by generalizing Hirota bilinear operators, and explore when the linear superposition principle can apply to the resulting generalized bilinear differential equations. Together with an algorithm using weights, two examples of generalized bilinear differential equations are computed to shed light on the presented general scheme for constructi...

Dahlberg’s Bilinear Estimate for Solutions of Divergence Form Complex Elliptic Equations

Ω ∇u · v, where u is harmonic in the domain Ω ≡ {(x, t) ∈ Rn+1 : t > φ(x)}, with φ Lipschitz, and where v ∈ W loc is vector valued. He showed that the bilinear form (1.1) is bounded by the L2 norm of the square function plus the non-tangential maximal function of u, times the same expression for v. In the present note, we generalize Dahlberg’s Theorem to variable coefficient divergence form ell...