Some Isoperimetric Inequalities in the Plane with Radial Power Weights

Sobolev and Isoperimetric Inequalities with Monomial Weights

We consider the monomial weight |x1|1 · · · |xn|n in R, where Ai ≥ 0 is a real number for each i = 1, ..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|1 · · · |xn|ndx, and they contain the best or critical exponent (which depends on A1, ..., An). More i...

Reverse Isoperimetric Inequalities In The Plane

Some inequalities related to isoperimetric inequalities with partial free boundary

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative function under Schwarz equimeasurable rearrangement. These ingredients also ...

Isoperimetric inequalities in the Heisenberg group and in the plane

We formulate the isoperimetric problem for the class of C2 smooth cylindrically symmetric surfaces in the Heisenberg group in terms of Legendrian foliations. The known results for the sub-Riemannian isoperimetric problem yield a new isoperimetric inequality in the plane: For any strictly convex, C2 loop γ ∈ R2, bounding a planar region ω, we have

Euclidean Balls Solve Some Isoperimetric Problems with Nonradial Weights

In this note we present the solution of some isoperimetric problems in open convex cones of R in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition i...