Sharp Sobolev inequalities in the presence of a twist

Abstract

Abstract. Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Let also A be a smooth symmetrical positive (0, 2)-tensor field in M . By the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where H 2 1 (M ) is the standard Sobolev space of functions in L 2 with one derivative in L 2 . We investigate in this paper the value of the sharp K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Also let A be a smooth symmetrical (0, 2)-tensor field in M . In a local chart, A = (A ij ), i, j = 1, . . . , n. We assume that A is positive when acting on 1-forms in the sense that for any x ∈ M , and any η in the cotangent spaceThen, by the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where ∇u = (∂ i u) is the 1-form consisting (in local charts) of the first derivatives of u, dv g is the Riemannian volume element of g, and H 2 1 (M ) is the standard Sobolev space consisting of functions in L 2 with one derivative in L 2 . Clearly, the sharp constant B in (0.1) is V −2/n g , where V g is the volume of (M, g), and the corresponding sharp inequality holds true since it holds true for the classical Sobolev inequality [and |∇u| 2 is controled by A x (∇u, ∇u)]. On the other hand, as is easily understood by the fact that A charges some parts of the space M more than others, it is expected that A will affect the sharp constant K in (0.1). Note (0.1) is associated to the operator ∆ g A u = −div g (A x ∇u) which appears in several places in mathematical and physics literature.The questions we ask in this note are: what is the value K s = K s (g) of the sharp K in (0.1), does the corresponding sharp inequality hold true, and if yes, does its saturated version (where B is lowered to its minimum value under the constraint K = K s ) possess extremal functions? When A = g −1 , we are back to the classical problem (dealing with the classical Sobolev inequality). Possible references in book

Extremal maps in best constants vector theory. Part I: Duality and compactness

“…In a natural way, an extremal map to (11) is defined as a non-zero k-map that realizes equality in (11). The adequate space to search extremal maps is the Riemannian k-vector Sobolev space…”

“…A successful best constants theory on sharp potential type Riemannian Sobolev inequalities relies on answers to a number of questions related to the optimal constants A 0 (n, F, G, g) and B 0 (n, F, G, g) and to the sharp Sobolev inequalities (10) and (11). In the same spirit of the scalar AB program, we inquire: (A) Is it possible to find the explicit values and/or bounds for A 0 (n, F, G, g) and B 0 (n, F, G, g)?…”

“…(C) Do A 0 (n, F, G, g) and B 0 (n, F, G, g) depend continuously on F , G and g in some topology? (D) Does the sharp potential type L 2 -Sobolev inequality (11) admit any extremal map? (E) Is the set of F -normalized extremal maps to (11) compact in some topology?…”

“…(D) Does the sharp potential type L 2 -Sobolev inequality (11) admit any extremal map? (E) Is the set of F -normalized extremal maps to (11) compact in some topology? A map U is said to be…”

“…for some constant λ > 0. Nevertheless, extremal maps to (11) do not need to be smooth and much less satisfying any equations, unless F (t) and G(x, t) are of C 1 class on t. In this specific situation, an extremal map U = (u 1 , . .…”