Eigenvalue estimates and localization of the first Dirichlet eigenfunction Thomas Beck University of North Carolina - Chapel Hill tdbeck@email.unc.edu April 3, 2020 Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 1 / 9
First Dirichlet eigenfunction u first Dirichlet eigenfunction, with eigenvalue λ : � (∆ + λ ) u = 0 in Ω , u = 0 on ∂ Ω . Throughout, Ω ⊂ R n will be a convex domain of inner radius 1. u = 0 on ∂ Ω, and we normalize u > 0 inside Ω. Superlevel sets: Ω c := { x ∈ Ω : u ( x ) > c } . Question How small a subset of the domain Ω can the eigenfunction localize to? � u � L 2 (Ω) Aim: Study localization via the quantity . � u � L ∞ (Ω) Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 2 / 9
An estimate of Chiti Theorem (Chiti ’82) � u � L 2 (Ω) ≥ c ∗ There exists a constant c ∗ n (independent of Ω ) such that n . � u � L ∞ (Ω) Equality holds precisely if Ω is the ball of radius 1 . As the diameter of Ω increases, is the eigenfunction forced to spread out along the diameter? Does there exist α > 0 (independent of the domain Ω) such that � u � L 2 (Ω) ≥ c n diam(Ω) α . � u � L ∞ (Ω) Conjecture (van den Berg ’00) There exists a constant c n (independent of the domain Ω ) such that � u � L 2 (Ω) ≥ c n diam (Ω) 1 / 6 . � u � L ∞ (Ω) Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 3 / 9
The two dimensional case Two explicit examples: 1) Rectangles, Ω = [0 , N ] × [0 , 1]: Solve via separation of variables, and � π x � u ( x , y ) = sin sin( π y ) . N � u � L 2 (Ω) is comparable to N 1 / 2 = diam(Ω) 1 / 2 . � u � L ∞ (Ω) 2) Circular sectors, Ω = { ( r , θ ) : 0 ≤ r ≤ α 1 ( N ) , 0 ≤ θ ≤ 1 N } : Solve via separation of variables, and u ( x , y ) = J N ( r ) sin ( π N θ ) . J N is the N th Bessel function, α 1 ( N ) ∼ N is its first zero. � u � L 2 (Ω) is comparable to N 1 / 6 = diam(Ω) 1 / 6 . � u � L ∞ (Ω) Georgiev-Mukherjee-Steinerberger ’18 showed that the sector is the most localized case and proved the conjecture in two dimensions, using work of Grieser and Jerison, ’95, ’96, ’98. Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 4 / 9
Localization of the eigenfunction Theorem (B. ’19) � u � L 2 (Ω) ≥ c n diam (Ω) 1 / 6 . There exists c n > 0 (independent of Ω ) such that � u � L ∞ (Ω) The 1 6 power cannot be improved in any dimension (attained by the cone). When Ω extends in more than one direction a stronger version of the inequality holds: Let K be a John ellipsoid for Ω, with axes lengths N 1 ≥ N 2 ≥ · · · ≥ N n ∼ 1 . Then, n − 1 � u � L 2 (Ω) N 1 / 6 � ≥ c n . j � u � L ∞ (Ω) j =1 Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 5 / 9
Localization of the eigenfunction The crucial estimate in the proof: Proposition There exists a constant C n (independent of Ω ) such that ˆ ˆ | ∂ x 1 u | 2 ≤ C n diam (Ω) − 2 / 3 u 2 . Ω Ω The theorem follows from this proposition combined crucially with the convexity of the superlevel sets (Brascamp and Lieb ’76). Idea of the proof of the proposition: 1) Reduce to eigenvalue bounds by writing x = ( x 1 , x ′ ) and combining: ˆ ˆ ˆ ˆ ˆ | ∂ x 1 u | 2 + |∇ x ′ u | 2 = λ |∇ x ′ u | 2 ≥ µ ∗ u 2 , u 2 . Ω Ω Ω Ω Ω Here µ ∗ is the minimal first eigenvalue of a ( n − 1)-dimensional cross-section of Ω, denoted by Ω ∗ . Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 6 / 9
Localization of the eigenfunction Proposition There exists a constant C n (independent of Ω ) such that ˆ ˆ | ∂ x 1 u | 2 ≤ C n diam (Ω) − 2 / 3 u 2 . Ω Ω ˆ ˆ | ∂ x 1 u | 2 ≤ ( λ − µ ∗ ) u 2 , and it remains to estimate 1) Combining implies Ω Ω λ − µ ∗ from above. 2) Do this by showing µ ∗ ≤ λ ≤ µ ∗ + C n diam(Ω) − 2 / 3 using the variational formulation of the first eigenvalue. Let ψ ∗ ( x ′ ) be the first Dirichlet eigenfunction of Ω ∗ , eigenvalue µ ∗ . Use the test function w ( x 1 , x ′ ) = χ ( x 1 ) ψ ( x ′ N 1 / ( N 1 − x 1 )) , with χ ( x 1 ) supported in an interval of length N 1 / 3 ∼ diam(Ω) 1 / 3 around Ω ∗ . 1 Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 7 / 9
Future Directions Theorem (B. ’19) � u � L 2 (Ω) ≥ c n diam (Ω) 1 / 6 . There exists c n > 0 (independent of Ω ) such that � u � L ∞ (Ω) � u � L 2 (Ω) ≤ vol(Ω) 1 / 2 . There is also the trivial upper bound � u � L ∞ (Ω) Question Can we use the geometry of the domain Ω to determine comparable upper and � u � L 2 (Ω) lower bounds on ? � u � L ∞ (Ω) In 2 dimensions, Jerison (’95), introduced a length scale L = L (Ω) to give a positive answer to this question. Open in higher dimensions. Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 8 / 9
Thank you for your attention! Thomas Beck (UNC) Localization of the first Dirichlet eigenfunction April 3, 2020 9 / 9
Recommend
More recommend
