On the Maxwell and Friedrichs/Poincaré constants in ND. (English) Zbl 1426.35007
Summary: We prove that for bounded and convex domains in arbitrary dimensions, the Maxwell constants are bounded from below and above by Friedrichs’ and Poincaré’s constants, respectively. Especially, the second positive Maxwell eigenvalues in ND are bounded from below by the square root of the second Neumann-Laplace eigenvalue.
MSC:
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35Q61 | Maxwell equations |
| 35E10 | Convexity properties of solutions to PDEs with constant coefficients |
| 35F15 | Boundary value problems for linear first-order PDEs |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
| 46E40 | Spaces of vector- and operator-valued functions |
| 53A45 | Differential geometric aspects in vector and tensor analysis |
| 35Q30 | Navier-Stokes equations |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 35J20 | Variational methods for second-order elliptic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
second Maxwell eigenvalue; electro statics; magneto statics; Poincaré inequality; Friedrichs inequality; Poincaré constant; Friedrichs constantReferences:
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