Hodge decompositions with mixed boundary conditions and applications to partial differential equations on lipschitz manifolds

On the Maxwell constants in 3D

“….q C 1/-forms. More precisely, Both latter properties of , that is, the approximation and the compactness property, hold, for example, if the boundary manifolds , t , n are Lipschitz and the boundary manifolds t , n are separated by a .N 2/-dimensional Riemannian and Lipschitz sub-manifold, the interface :D t \ n , see [23,24] for details and proofs. We note that…”

Exit Event from a Metastable State and Eyring-Kramers Law for the Overdamped Langevin Dynamics

“…L 2 w stands for both the scalar product associated with the norm of L 2 w (Ω) and with the norm of Λ 1 L 2 w (Ω). In view of (29) and (28), one has for all orthonormal basis (ψ j ) j∈{1,...,n} of Ran π…”

“…with mixed boundary conditions onΩ j , the recent results of [38] and [28] are used. The 1-form ψ j associated with z j is then defined using an eigenform v (1) h,j associated with the eigenvalue 0 of the operator L (1) f,h associated with mixed boundary conditions onΩ j :…”

Smoothed projections and mixed boundary conditions

“…Special cases are the Poisson equation with mixed Dirichlet and Neumann boundary conditions [46], and the vector Laplace equation with mixed tangential and normal boundary conditions [26]. It is known in the theory of partial differential equations that the Hodge Laplace equation with mixed boundary conditions arises from Sobolev de Rham complexes with partial boundary conditions [32,27]. These are composed of spaces of Sobolev differential forms in which boundary conditions are imposed only on a part of the boundary (corresponding to the essential boundary conditions).…”

“…The additional complexity in comparison to the scalar-valued case begins with the correct definition of tangential and normal boundary conditions in a setting of low regularity [52,12,13,53,27]. When non-mixed boundary conditions are imposed, so that either Γ T = ∅ or Γ T = ∂Ω, then Rellich-type compact embeddings H(div, Ω, Γ N ) ∩ H(curl, Ω, Γ T ) → L 2 (Ω, R 3 ) and vector-valued Poincaré-Friedrichs inequalities have been known for a long time [51,47,55,17].…”