Best Constant in Sobolev Inequality
نویسنده
چکیده
The equality sign holds in (1) i] u has the Jorm: (3) u(x) = [a + btxI,~',-'] 1-~1~ , where Ix[ = (x~ @ ...-~x~) 1⁄2 and a, b are positive constants. Sobolev inequalities, also called Sobolev imbedding theorems, are very popular among writers in part ial differential equations or in the calculus of variations, and have been investigated by a great number of authors. Nevertheless there is a question concerning Sobolev inequalities, which seems well-known only to a restricted number of specialists working in geometric measure theory. The question is the connection between Sobolev inequalities and the classical isoperimetrie inequality for subsets of euclidean spaces. Our aim is to advertise such a connection. To be specific, we are concerned with the simplest Sobolev inequality (~) ][ u [I L~(~ m) < (constant independent of u)]1Du 11L~(~m),
منابع مشابه
Logarithmic Sobolev Trace Inequality
A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality. Logarithmic Sobolev inequalities capture the spirit of classical Sobolev inequalities with the logarithm function replacing powers, and they can be considered as limiting cases of...
متن کاملOn the Best Sobolev Inequality
We prove that the best constant in the Sobolev inequality (WI,” c Lp* with $= f i and 1 c p < n) is achieved on compact Riemannian manifolds, or only complete under some hypotheses. We also establish stronger inequalities where the norms are to some exponent which seems optimal. 0 Elsevier, Paris
متن کاملGlobal Poincaré Inequalities on the Heisenberg Group and Applications
Let f be in the localized nonisotropic Sobolev space W 1,p loc (H ) on the n-dimensional Heisenberg group H = C × R, where 1 ≤ p < Q and Q = 2n + 2 is the homogeneous dimension of H. Suppose that the subelliptic gradient is gloablly L integrable, i.e., Hn |∇Hnf |pdu is finite. We prove a Poincaré inequality for f on the entire space H. Using this inequality we prove that the function f subtract...
متن کاملLooking for the Best Constant in a Sobolev Inequality: A Numerical Approach
A numerical method for the computation of the best constant in a Sobolev inequality involving the spacesH2(Ω) and C0(Ω) is presented. Green’s functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagr...
متن کاملExtremals for the Sobolev Inequality on the Seven Dimensional Quaternionic Heisenberg Group and the Quaternionic Contact Yamabe Problem
A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the L Folland-Stein embedding theorem is determined.
متن کامل