Isoperimetric inequalities for eigenvalues of triangles

Isoperimetric inequalities for eigenvalues oftriangles

Bartłomiej Siudeja

Purdue University

19 September 2007

Bartłomiej Siudeja (Purdue University) Isoperimetric inequalities for eigenvalues 19 September 2007 1 / 15

Outline

1 IntroductionNotation and classical resultsNew results

2 Symmetrization techniquesSteiner symmetrizationContinuous Steiner symmetrizationPolarization

3 ProofsFreitas’s lower boundImproved lower boundIsosceles trianglesCircular sectorsMonotonicity

Introduction Notation and classical results

EigenvaluesLet D be an open set. Eigenvalues λi of the Dirichlet Laplacian on Dwill be called eigenvalues of D. They form a nondecreasing sequencesuch that 0 < λ1 < λ2.

Geometric quantitiesA - area of DR - inradiusL - perimeterd - diameterλD = λ1 - first eigenvalue of D

Trianglesh - altitude perpendicular to the longest sideγ - smallest angle

EigenvaluesLet D be an open set. Eigenvalues λi of the Dirichlet Laplacian on Dwill be called eigenvalues of D. They form a nondecreasing sequencesuch that 0 < λ1 < λ2.

Geometric quantitiesA - area of DR - inradiusL - perimeterd - diameterλD = λ1 - first eigenvalue of D

Trianglesh - altitude perpendicular to the longest sideγ - smallest angle

Classical isoperimetric inequalityAmong all domains with fixed area A, the ball minimizes perimeter L.

L2 ≥ 4πA.

Theorem (Faber-Krahn inequality)Among all domains with fixed area A, the ball minimizes the firsteigenvalue.

Pólya’s isoperimetric conjectureAmong all polygons P(n) with n sides and fixed area A, the regularpolygon R(n) minimizes the first eigenvalue.

λP(n)A ≥ λR(n)A.

(Proved for triangles and quadrilaterals.)

L2 ≥ 4πA.

Known eigenvaluesballrectanglesannulicircular sectorsequilateral triangleright triangles with angles π/4 or π/6

Methods of obtaining lower boundsDomain monotonicity (larger domain→ smaller eigenvalue)Restricting to a subdomain (right isosceles triangle is a half of asquare)Special analytical casesSymmetrization

Known eigenvaluesballrectanglesannulicircular sectorsequilateral triangleright triangles with angles π/4 or π/6

Methods of obtaining lower boundsDomain monotonicity (larger domain→ smaller eigenvalue)Restricting to a subdomain (right isosceles triangle is a half of asquare)Special analytical casesSymmetrization

Introduction New results

Theorem (Freitas ’06)For arbitrary triangle T

λT ≥ π2(

Theorem

λT ≥ π2(

4d2 + h2 +

d2 + h2

TheoremLet T be a triangle with fixed area A and smallest angle γ. Then theeigenvalue λT is decreasing with diameter d.

λT ≥ π2(

Theorem

λT ≥ π2(

4d2 + h2 +

d2 + h2

λT ≥ π2(

Theorem

λT ≥ π2(

4d2 + h2 +

d2 + h2

TheoremThe eigenvalue λT of an isosceles triangle T with area A and smallestangle γ is bigger then the eigenvalue of a sector with the same areaand angle.

TheoremGiven fixed area A, the eigenvalue of an isosceles triangle decreaseswhen the smallest angle γ increases.

The same is true for righttriangles.

TheoremGiven fixed area A, the eigenvalue of an isosceles triangle decreaseswhen the smallest angle γ increases.

The same is true for righttriangles.

TheoremGiven fixed area A, the eigenvalue of an isosceles triangle decreaseswhen the smallest angle γ increases. The same is true for righttriangles.

Symmetrization techniques Steiner symmetrization

Definition of Steiner symmetrizationFix a line l and domain D.

Consider cross-sections of a domain Dperpendicular to l . The Steiner symmetrization of D is the domain D∗

formed by the cross-sections centered around l .

Action on triangles

preserves areadecreases perimeterdecreases the first eigenvalue

Definition of Steiner symmetrizationFix a line l and domain D. Consider cross-sections of a domain Dperpendicular to l .

The Steiner symmetrization of D is the domain D∗

Action on triangles

Definition of Steiner symmetrizationFix a line l and domain D. Consider cross-sections of a domain Dperpendicular to l . The Steiner symmetrization of D is the domain D∗

Action on triangles

Symmetrization techniques Continuous Steiner symmetrization

Definition of continuous Steiner symmetrization

Let 0 ≤ t ≤ 1 be a time parameter. Let D0 equal to initial domain D,and D1 equal to the Steiner symmetrization D∗.

For 0 < t < 1 wedefine a continuous Steiner symmetrization as a domain formed by thecross-sections at time t (partially shifted proportionally to t).

Action on triangles

D = D0 D1 = D∗

As t increases:area remains fixedperimeter decreasesthe first eigenvaluedecreases

Let 0 ≤ t ≤ 1 be a time parameter. Let D0 equal to initial domain D,and D1 equal to the Steiner symmetrization D∗. For 0 < t < 1 wedefine a continuous Steiner symmetrization as a domain formed by thecross-sections at time t (partially shifted proportionally to t).

Action on triangles

DtD = D0 D1 = D∗

As t increases:area remains fixedperimeter decreasesthe first eigenvaluedecreases

Let 0 ≤ t ≤ 1 be a time parameter. Let D0 equal to initial domain D,and D1 equal to the Steiner symmetrization D∗. For 0 < t < 1 wedefine a continuous Steiner symmetrization as a domain formed by thecross-sections at time t (partially shifted proportionally to t).

Action on triangles

DtD = D0 D1 = D∗ As t increases:area remains fixedperimeter decreasesthe first eigenvaluedecreases

Symmetrization techniques Polarization

Definition of polarizationFix a domain D and a line l . This line splits the space into twohalf-spaces H1 and H2. Let D′ be a reflection of D about l . Thepolarization DP of a set D consists of D ∩ H1, D′ ∩ H1 and any pointfrom (D ∪ D′) ∩ H2 such that its reflection is already included.

Action on triangles

preserves areadecreases perimeterdecreases the firsteigenvalue

Action on triangles

Proofs Freitas’s lower bound

Alternative proof of Freitas’s bound

l2αα

Steps1. Steiner symmetrization with respect to l1.

l2αα

Steps2. Polarization with respect to l2.

l2αα

Steps4. The same procedure on the other side

l2αα

5. λT ≥ λR = π2(

1a2 + 1

)= π2

d2 + d2

)Bartłomiej Siudeja (Purdue University) Isoperimetric inequalities for eigenvalues 19 September 2007 11 / 15

Proofs Improved lower bound

Improved lower bound using rectangles

StepsStart with an already symmetrized triangle.

Steiner symmetrization with respect to the longest side.Steiner symmetrization with respect to the altitude.

λT ≥ λR = π2(

1a2 + 1

)= π2

d2+h2 + d2+h2

StepsStart with an already symmetrized triangle.Steiner symmetrization with respect to the longest side.

Steiner symmetrization with respect to the altitude.

λT ≥ λR = π2(

1a2 + 1

)= π2

d2+h2 + d2+h2

StepsStart with an already symmetrized triangle.Steiner symmetrization with respect to the longest side.Steiner symmetrization with respect to the altitude.

λT ≥ λR = π2(

1a2 + 1

)= π2

d2+h2 + d2+h2

StepsStart with an already symmetrized triangle.Steiner symmetrization with respect to the longest side.Steiner symmetrization with respect to the altitude.

λT ≥ λR = π2(

1a2 + 1

)= π2

d2+h2 + d2+h2

)Bartłomiej Siudeja (Purdue University) Isoperimetric inequalities for eigenvalues 19 September 2007 12 / 15

Proofs Isosceles triangles

Symmetrization into isosceles triangles

We want to fit a gray triangle with area A inside a red one with areaA + ε. Then by the scaling property we can take a limit ε→ 0. Thisshows that given area A and the smallest angle γ, the eigenvaluedecreases with diameter.

We need to define a sequence of polarizations with respect to certainbisectors.

The reversed sequence of reflections gives a valid sequence ofpolarizations.

Proofs Circular sectors

Symmetrization into circular sector

Proofs Monotonicity

Monotonicity for isosceles triangles

Continuous Steiner symmetrization with respect to l1.Continuous Steiner symmetrization with respect to l2.

Proofs Monotonicity

Continuous Steiner symmetrization with respect to l1.

Continuous Steiner symmetrization with respect to l2.

Proofs Monotonicity

Continuous Steiner symmetrization with respect to l1.Continuous Steiner symmetrization with respect to l2.

Isoperimetric inequalities for eigenvalues of triangles

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