Evaluation of the Least Constant in Sobolev’s Inequality for $H^1 (0,s)$

Marti, Jürg T.

doi:10.1137/0720094

Cited by 27 publications

(20 citation statements)

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“…While Sobolev space embeddings are of general importance in the study of partial differential equations, their application in the context of computer-assisted proofs for nonlinear problems relies crucially on the explicit knowledge of the associated embedding constants. Even though there are results for general Sobolev spaces on one-dimensional domains, [28][29][30] as well as for Sobolev spaces on higher-dimensional domains subject to homogeneous Dirichlet boundary conditions, 11 in the context of problems subject to homogeneous Neumann boundary conditions, these constants cannot easily be found in the literature. This is true even for the important case of higher-dimensional rectangular domains.…”

Section: Discussionmentioning

confidence: 99%

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Validated bounds on embedding constants for Sobolev space Banach algebras

Wanner

2018

Math Methods in App Sciences

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MSC Classification: 46E35; 65G20; 65G30; 35G30; 46J15 Sobolev spaces and their embedding properties have long been of central importance in the study of partial differential equations, particularly for the classical theoretical analysis of both linear and nonlinear problems. In recent years, computer-assisted proof techniques have been developed for obtaining existence and uniqueness proofs of solutions to a variety of nonlinear partial differential equations that arise in applications, and they have led to a number of new results that currently lie beyond the reach of classical analytical approaches.The use of computer-assisted methods, however, frequently relies on the explicit knowledge of a variety of embedding constants for Sobolev spaces. In the present paper, we show that in the context of certain Sobolev space Banach algebras, these constants themselves can be bounded rigorously and precisely using validated computations based on interval arithmetic, combined with analytical error estimates. KEYWORDSBanach algebra, computer-assisted proof, embedding constant, embedding theorems, interval arithmetic, Sobolev space Math Meth Appl Sci. 2018;41:9361-9376.wileyonlinelibrary.com/journal/mma

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Section: Discussionmentioning

confidence: 99%

“…Together with (28), this implies that we only need to determine upper bounds for the values f(k) for all k ∈ N 3 0 with 0 ≤ k 1 ≤ k 2 ≤ k 3 and |k| ≤ 200. This reduces the number of bounds that have to be derived using interval arithmetic by more than one order of magnitude.…”

Section: Banach Algebra Estimates: Reduction and Final Estimatesmentioning

confidence: 99%

Validated bounds on embedding constants for Sobolev space Banach algebras

Wanner

2018

Math Methods in App Sciences

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“…We show that this constant is equal to the norm of the general Sobolev imbedding operator on the interval [0, rc] which was calculated by Marti [10]. In the remaining sections we discuss the relationship between 6w and the widely examined Hausdorff metric tS~.…”

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confidence: 79%

On the Sobolev distance of convex bodies

Arnold

Wellerding

1992

Aeq. Math.

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On the Sobolev distance of convex bodies RANDOLF ARNOLD AND ANDREAS WELLERDINGSummary. Let 5~ rd denote the cone of all convex bodies in the Euclidean space E a. The mapping K~--~h x of each body K e.X¢ "a onto its support function induces a metric 6w on oF d by 6~(K, L):= ]lhL--hxl ] w where rl" IIw is the Sobolev l-norm on the unit sphere 5a-lc ~a. We call 6w(K, L) the Sobolev distance of K and L. The goat of our paper is to develop some fundamental properties of the Sobolev distance.

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“…Although, Hegland and Marti [3,Theorem 5] show this fact for j = 0, the proof (essentially) does not seem applicable to our case j 1. Nevertheless, the method developed in Marti [4] which considers the simplest case m = 1 and j = 0 applies to our case. To follow the argument of [4], we first compute K (j ) j (0, 0).…”

Section: Proofs Of Theoremsmentioning

confidence: 97%

“…Nevertheless, the method developed in Marti [4] which considers the simplest case m = 1 and j = 0 applies to our case. To follow the argument of [4], we first compute K (j ) j (0, 0). As in the case of a = ∞, K j (·, 0) is given as a function satisfying (10) and boundary conditions…”

Section: Proofs Of Theoremsmentioning

confidence: 97%