Sobolev spaces of vector-valued functions. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 190(Issled. po Linein. Oper. i Teor. Funktsii. 19):5-14, 185, 1991.Bukhvalov, A.V.: Sobolev spaces of vector-valued functions. Zap. Nauchn. Sem. Leningrad. Otdel. ...
Finally, we provide a chain rule for the truncation of vector-valued Sobolev functions. Similar to the classical argument in the scalar-valued case, we use a smooth approximation of the truncation and pass to the limit by using the dominated convergence theorem....
Our definitions are meaningful for vector valued functions on general measure metric spaces as well and seem to lead to the most natural class of metric Sobolev spaces. The discussion of higher order Sobolev spaces and higher order mean difference quotients on regular subsets of Euclidean spaces is...
Also, we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it is acting as a superposition operator. 关键词: Sobolev spaces Banach function space Vector-valued functions Reshetnyak-Sobolev space Newtonian space DOI: 10.1016/j.na....
Briggs_Calculus_11_Vectors_and_Vector_Valued_Functions_split_5 Briggs_Calculus_11_Vectors_and_Vector_Valued_Functions_split_1 Semicontinuity Results on Parametric Vector Variational Inequalities with Polyhedral Constraint Sets On the computation of an element of Clarke generalized Jacobian for a vector-va...
We may define the operator that values the "Energy" of the function Definition 1.Energy OperatorJ(u)is defined as J(u)=∫B(12|∇u|2+f(x)u(x))dx We may be interested in the minimum of the operator, ˉu=argminu∈H10(B)J(u) ...
1.In this paper, we discuss the global bifurcation result for the p — harmonic operatorin the weighted Sobolev space with Navier boundary conditionDenoteFor any , we definewhere w(x) = {wi(x)}_i~n=0 are vector-valued functions, and W_0~(1,p)(Ω,w) denotes the weighted Sobolev sp...
a measurable vector-field valued function, which we again denote by f, we can take its length in the Riemannian metric, | f| x := f, f x , and (dropping the subscript x for simplicity) compute the semi-norms f p := M | f| p dµ 1/p , p ≥ 1. The nonhomogeneous ...
(of all orders) are continuous on.shall denote those functionsfor which there exists a compact setsuch thatfor. When a function is vector-valued we shall instead write,, or. For,shall denote the usual Sobolev space of (Lebesgue) measurable (vector-valued) functionswhose distributional gradient...
We can now extend this to the setting of general vector-valued sequence spaces of type \\ell_q(\\beta_j \\ell_p^{M_j}) \\ell_q(\\beta_j \\ell_p^{M_j}) with 1\\leq p,q\\leq\\infty 1\\leq p,q\\leq\\infty , M_j\\in {\\mathbb N}_0 M_j\\in {\\mathbb N}...