Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities

We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin.

     

Extremal functions for the Caffarelli-Kohn-Nirenberg inequalities: A simple proof of the symmetry

In this article, we give a simple proof of the result due to Lin and Wang ensuring the foliated Schwarz symmetry of the extremal functions for the Caffarelli-Kohn-Nirenberg inequalities. This new proof uses a direct and powerful method due to Bartsch, Weth and Willem using polarizations.     

A sharp estimate for extremal functions

We prove a sharp pointwise estimate for extremal functions of invariant subspaces of some weighted Bergman spaces on the unit disk. The allowed weights include standard radial weights and logarithmically subharmonic weights.

     

Bergman space extremal problems for non-vanishing functions

We consider the problem of minimising the Bergman Space A ² norm of functions analytic and non-vanishing in the unit disc, which satisfy a finite number of constraints of the form l _i (f) = c _i , where each l _i (f) is a finite linear combination of Taylor coefficients of f evaluated at certain points of the disc. We show that when the class of functions satisfying the constraints is nonempty, an extremal function exists and that every extremal function has rational outer part of a specific form.     

A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$ for some pointz ⁰∈ω, where $$h_E (z) = \sup \{ u(z):u \in PSH (\Omega ),u \leqslant 0 on E,u \leqslant 1 in \Omega \} ,z \in \Omega ,$$ is the extremal function often studied in complex analysis.     

Existence of extremal functions for a Hardy-Sobolev inequality

We present the best constant and the extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in RN.     

Some extremal bounds for subclasses of univalent functions

In this paper we obtain the sharp lower bound for , for functions f that are k-uniformly convex in the unit disk U. Next we consider the problem of finding the minimum of for functions f that are k-uniformly convex in the disk of radius r. Corresponding results for the class of starlike functions related to the class of k-uniformly convex functions are presented.     

Asymptotics for the best Sobolev constants and their extremal functions

Let , where Ω is a bounded domain of , , and . We prove that , where ρ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of converge (as ) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured domain , for some .     

On extremal points of quasiconvex functions

In this paper we examine the linear sectionwise relative minimums of a quasiconvex function and give a sufficient condition for quasiconvex functions to have a strict global minimum on an open convex set.     

An extremal property of outer functions

Our main result is the following: iff (z) is in the space H², and F(z) is its outer part, then F⁽ⁿ⁾_H2F⁽ⁿ⁾_H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971.     

An extremal problem related to Kolmogoroff’s inequality for bounded functions

LetA andB be positive numbers andm andn positive integers,m. Then there is for complex valued functions φ onR with sufficient differentiability and boundedness properties a representation wherev ₁ andv ₂ are bounded Borel measures withv ₁ absolutely continuous, such that there exists a function φ with ∣φ⁽ⁿ⁾∣ ?A and ∣φ∣ ?A onR and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.