Open Subset Of Rn Research Articles

An approximation theorem of Runge type for the heat equation

If Ω \Omega is an open subset of R n + 1 {{\mathbf {R}}^{n + 1}} , the approximation problem is to decide whether every solution of the heat equation on Ω \Omega can be approximated by solutions defined on all of R n + 1 {{\mathbf {R}}^{n + 1}} . The necessary and sufficient condition on Ω \Omega which insures this type of approximation is that every section of Ω \Omega taken by hyperplanes orthogonal to the t t -axis be an open set without “holes,” i.e., whose complement has no compact component. Part of the proof involves the Tychonoff counterexample for the initial value problem.

�ber eine Koerzitivit�tsungleichung in W o m,p

Let G be a bounded open subset of Rn and let B be a Dirichlet bilinear form with bounded coefficients defined in G. If the generalized version of Garding's inequality in Lp holds for all u of the Sobolev space W0m,p(G), then we prove under additional assumptions that the form is uniformly elliptic in G and that for n=2 a root condition is satisfied. Further, we study the equivalence of some definitions of “properly elliptic”.

Functions possessing restricted mean value properties

A real-valued function f f defined on an open subset of R N {R^N} is said to have the restricted mean value property with respect to balls (spheres) if, for each point x x in the set, there exists a ball (sphere) with center x x and radius r ( x ) r(x) such that the average value of f f over the ball (sphere) is equal to f ( x ) f(x) . If f f is harmonic then it has the restricted mean value property. Here new conditions for the converse implication are given.

Open mappings and the lack of full completeness of 𝒟’(Ω)

Consider a linear map T T of one locally convex linear space into another which is densely defined and has a closed graph. We characterise the property that T T is an open map in terms of two properties of its adjoint map T ∗ {T^{\ast }} . These results are used to show that if Ω \Omega is an open subset of R n {R_n} for which there is a linear constant coefficient differential operator P P such that Ω \Omega is P P -convex but not strongly P P -convex, then (i) D ′ ( Ω ) \mathcal {D}’(\Omega ) is not fully complete, (ii) the range of the adjoint map t P ^tP is closed but not bornological.