On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets.
Let be an open convex set in with finite width, and let be the torsion function for , i.e. the solution of . An upper bound is obtained for the product of , where is the bottom of the spectrum of the Dirichlet Laplacian acting in . The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area , it is shown that , and that this bound is sharp.
AMS 2000 subject classifications. 49J45, 49R05, 35P15, 47A75, 35J25.
Key words and phrases. torsion function, torsional rigidity, first Dirichlet eigenvalue
Let be an open set in , and denote the bottom of the spectrum of the Dirichlet Laplacian acting in by
then the torsion function, defined by
In  it was shown that
The sharp constant in the right-hand side of (1.1) is not known. However, for an open ball , an open square , and an equilateral triangle ,
For open, bounded convex sets in it was shown in (3.12) of  that
with equality in the limit of an infinite slab (the open set with finite width bounded by two parallel -dimensional planes). The latter assertion has been made precise in  where it was shown that if
For bounded planar convex sets with width , and diameter , it was shown in  that
In Theorem 1.1 below we put (1.4)-(1.5), and (1.6) in a more general setting. We introduce the following notation. For an open, bounded, convex set with finite width , and boundary we let , be a family of parallel hyper-planes such that and are tangent to at two points and respectively, where , and is orthogonal to . That this is always possible was shown in Theorem 1.5 in . We identify sets in with sets in . Let
The projection of onto is denoted by
We denote the inradius of this -dimensional set by . The measure of is denoted by .
If is an open, bounded, convex set in then
The torsional rigidity (or torsion) of an open set is defined by
In Pólya and Szegö , it was shown that for sets with finite measure ,
It was subsequently shown in  that the constant in the right-hand side above is sharp: for , there exists an open, bounded, and connected set such that
The quantity in the left-hand side of (2) is invariant under the homothety transformation . This implies for example that in Theorems 1.2-1.5 below we do not have to specify the actual lengths of the edges of the rhombi and triangles. In the proofs of these theorems we fix the various lengths as a matter of convenience.
It was shown in Theorem 1.5 of  that for a thinning (collapsing) sequence of bounded convex sets
This supports the conjecture that for bounded, convex sets the sharp constant in the right-hand side of (2) is .
It was shown in Theorem 1.4 in  that for bounded convex sets in
and that for planar, bounded convex sets the inequality holds with constant . In  it was conjectured that for planar, bounded, convex sets
and that this constant is sharp for a thinning (collapsing) sequence of isosceles triangles. See also . By (1.3) and (1.6) above, we have that for a thin isosceles triangle . This suggests that the sharp constant for planar convex sets in the right-hand side of (1.12) is . We have the following.
If is an isosceles triangle with angles , and if then
If is a rhombus with angles , and if then
If is as in Theorem 1.3, then
If is an isosceles triangle with angles , then
This paper is organised as follows. In Section 2 we prove Theorem 1.1. The proofs of Theorems 1.2 and 1.3 are deferred to Section 3. The proof of Theorem 1.4 is deferred to Section 4. The proof of Theorem 1.5 consists of two parts. In Section 5 part 1 we show that inequality (1.17) holds for all , where
2 Proof of Theorem 1.1
Proof. We first observe, that by domain monotonicity of the torsion function, is bounded by the torsion function for the (connected) set bounded by and . Hence
It suffices to obtain an upper bound for . We choose the -coordinates such that . By convexity we have that the convex hull of is contained in . This convex hull in turn contains a cylinder with height , and base Denote the first dimensional Dirichlet eigenvalue of by . Then, by separation of variables, we have that
The right-hand side of (2.1) is minimised for
This gives that
Denote the inradius of , and the centre of the inball by and , respectively:
The inball intersects in a -dimensional disc with radius bounded from below by
where we have used that , see Blaschke’s theorem, p. 215 in . Hence
By convexity we have that . Hence is bounded from above by times the bottom of the spectrum of . The latter contains a -dimensional ball with radius . So
Proof of Theorem 1.2. Let be an isosceles triangle with a base of length and width (height) of length , and angles , and respectively. By hypothesis, so that . We denote the infinite sector with opening angle by
It is straightforward to verify that the torsion function for is given by,
We can cover with two sectors of opening angles and radii each. By monotonicity and positivity of the torsion function we have that
where we have used that . By adapting formula (31) in the proof of Theorem 2 in  to the geometry of we find that
which completes the proof of Theorem 1.2.
Proof of Theorem 1.3. Let be a rhombus with angles , and diagonals of length and respectively. By hypothesis we have that , and This rhombus is covered by two sectors of opening angle , and radius By the calculations in the proof of Theorem 1.2 we find that
By adapting formula (31) in the proof of Theorem 2 in  to the geometry of we find that
This, together with gives that,
This concludes the proof of Theorem 1.3.
4 Proof of Theorem 1.4
Let be a rhombus such that major and minor diagonals have lengths and , respectively (see Figure 1). We want to estimate the torsion and to this aim we use a test function
In view of the variational definition of the torsion we have
On the other hand we can estimate from below the first Dirichlet Laplacian eigenvalue of any rhombus by means of the Dirichlet Laplacian eigenvalue of a rectangle obtained by Steiner symmetrising the rhombus along a direction parallel to one of the sides (see Figure 2). We denote by and the base and the height of the rectangle, respectively. Since the base coincides with the side of the rhombus know that , the height .
We have that
Observing that the area of the rhombus is equal to , we have that
5 Proof of Theorem 1.5
5.1 Proof for the case
Let be an isosceles triangle with angles We first consider the case . We denote the height by , and we fix the length of the basis equal to . See Figure 3.
We use the function
as a test function for the torsion of . We find that
We wish to estimate from below. To this aim we consider the first Dirichlet eigenfunction of restricted to and we reflect it, anti-symmetrically, with respect to the line (see Figure 4). This new function is a test function defined on the rectangle of sides (shaded in grey in Figure 4) orthogonal to the first eigenfunction of the Laplacian with the mixed boundary conditions described in Figure 4.
Next we consider the case or . We have that
be the circular sector with radius and opening angle . Siudeja’s Theorem 1.3 in  asserts that for , , where is such that . It follows that
where is the first negative zero of the Airy function. It follows that
The torsion function for is given by (p.279 in ),
By monotonicity of the torsion we obtain that
We have that for , This gives that
The right-hand side of (5.5) is greater or equal than for
Inequality (5.11) holds for all
5.2 Computer validation for the case via interval arithmetic.
We consider a triangle of height and opening angle , where . Let
We wish to show that in the range .
We present here a computer assisted proof of the result using Interval Arithmetic.
We once more use Siudeja’s lower bound, comparing with the sector having the same opening angle and the same area (Theorem 1.3 of ), to get that
where is given by (5.7).
The area is given by
The monotonicity of with respect to inclusion allows us to estimate from below using the torsion of a tangent sector with same opening angle . We use (5.1) and find that
In order to perform a numerical evaluation we truncate the series in the following way
It follows that
At this point we can prove that for all values by using Interval Arithmetic. There are many softwares and libraries which can be employed for this purpose. We selected Octave111John W. Eaton, David Bateman, SÃ¸ren Hauberg, Rik Wehbring (2018). GNU Octave version 4.4.1 manual: A high-level interactive language for numerical computations. URL https://www.gnu.org/software/octave/doc/v4.4.1/ (A free software that runs on GNU/Linux, macOS, BSD, and Windows) which provides a specific package called Interval.222Oliver Heimlich, GNU Octave Interval Package, https://octave.sourceforge.io/interval/, version 3.2.0, 2018-07-01. The interval package is a collection of functions for interval arithmetic. It is developed at Octave Forge, a sibling of the GNU Octave project.
We covered the interval by a collection of 1001 intervals with , so that
We observe that the intersection of consecutive intervals is intentionally non empty.
Using the Interval package, we designed a code that for going from to provides upper and lower bounds for in terms of floating point numbers. This is performed in an automated way by standard and reliable algorithms. We established that the inequality holds true on the whole interval by verifying it on for all .
For completeness we include the code:
Acknowledgments. The authors wish to thank Prof. Gerardo Toraldo for helpful discussions on Interval Arithmetic. The authors acknowledge support by the London Mathematical Society, Scheme 4 Grant 41636. MvdB was also supported by The Leverhulme Trust through Emeritus Fellowship EM-2018-011-9, and by INDAM-GNAMPA Grant for visiting professors.
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