Randwertaufgaben in der verallgemeinerten Potentialtheorie

Abstract

The Neumann and Dirichlet boundary value problem of generalized potential theory is considered. Based on a compact imbedding result, existence and uniqueness theorems are obtained for Riemannian manifolds with compact boundary (‘interior and exterior domains’).

On Korn's first inequality for tangential or normal boundary conditions with explicit constants

“…This follows by the simple fact that or equivalently holds with , where the skew‐symmetric matrix corresponds to and the vector field to the vector field . An adequate solution theory for these electro‐magneto static problems can be found in . In fact, is the Helmholtz projection π of onto solenoidal vector fields with homogeneous normal boundary condition.…”

“…/ holds with Ȯ .x/ :D˙x, where the skew-symmetric matrix˙2 R N N corresponds to 2 R .N 1/N=2 and the vector field Ȯ to the vector field O . An adequate solution theory for these electro-magneto static problems can be found in [22][23][24][25][26]. In fact, v D O is the Helmholtz projection of O onto solenoidal vector fields with homogeneous normal boundary condition.…”

On the Maxwell constants in 3D

“…Another successful approach proving the Maxwell compactness property using a different technique from [2] has been shown in [3]. For the Maxwell compactness property in the case of full boundary conditions, we refer to [2,[4][5][6][7][8][9][10][11][12][13][14].…”

On a selfadjoint realization of curl in exterior domains

“…From this we have in particular the validity of the assumptions of Theorem 2.6. The needed closedness of curl • H (curl, Ω ∩ B(0, R)), R sufficiently large, is a special case of [16], Lemma 3. For the statement on the finite-dimensionality of the spaces of harmonic Neumann vector fields, see [16], Lemma 4 and Corollary 7.2.…”