Eigenfunctions for versions of the p-laplacian

Transcription

1 Eigenfunctions for versions of the p-laplacian Bernd Kawohl Universität zu Köln kawohl *Joint work with C. Nitsch, L.Esposito, C.Trombetti and others B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

2 Geometry of the Laplacian u = u x1 x u xnx n = u νν + u ν div(ν) where ν(x) = u(x) u(x) is direction of steepest descent. In fact, div(ν) = u u + u x i u xj u xi x j u 3 so that u = u νν u div(ν) = u νν + u ν div(ν) or u = u νν + u ν (n 1)H with H denoting mean curvature of a level set of u. For radial u recall u = u rr + n 1 r u r. = u u + u νν u B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

3 For p (1, ) one can write the p-laplace operator as p u = div ( u p 2 u ) = u p 2 [ u + (p 2)u νν ] = u p 2 [(p 1)u νν + (n 1)Hu ν ] and the normalized or game-theoretic p-laplace operator as N p u = 1 p u 2 p div ( u p 2 u ) = p 1 p u νν + 1 p (n 1)Hu ν = p 1 p N u + 1 p N 1 u. Observe N u = u νν, N 2 u = 1 2 u and N u 1 u = u div( u ). Parabolic counterparts are u t p u = 0 and u t N p u = 0 (Does, Banerjee). B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

4 Original and perturbed image B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

5 p = 1.2 and p = 10 after 5 identical time-steps B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

6 p = 1.2 and p = 10 after 15 identical time-steps B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

7 Parabolic counterparts are u t p u = 0 and u t N p u = 0 (Does, Banerjee). v t 1 v = v t (n 1)H v t N 1 v = 0 is known as TV flow and used in image processing. is the level set formulation of MC flow (Evans,Spruck). Ansatz v(t, x) = T (t)u(x) for u t + Au = 0 and A homogeneous of degree d leads to eigenvalue value problems Au = λu and a time decay T (t) which is exponential iff the eigenvalue problem is homogeneous of degree 1, as in Au = N p u or Au = u 2 p p u. This is how one arrives at the eigenvalue problems N p u = λu and p u = λ u p 2 u for p (1, ). B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

8 Consider the Dirichlet eigenvalue problem for the p-laplacian p u = λ p p u p 2 u in Ω u = 0 with first eigenvalue λ p p minimizing R p (v) := on Ω, Ω v p Ω v p on W 1,p 0 (Ω). (1) What is known about λ p and what happens as p? λ p λ := 1/R(Ω), where R(Ω)is the inradius. Moreover, up to a subsequence, the (nonnegative) eigenfunctions v p converge uniformly to a positive Lipschitzfunction v, solving min{ v λ v, v} = 0 in Ω, v = 0 on Ω in the sense of viscosity solutions. (Juutinen, Lindqvist, Manfredi 99) B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

9 Here v = n i,j=1 v i v ij v j = v 2 v νν. Note that the limiting equation for p u = λ p p u p 2 is min{ v λv, v} = 0 in Ω, v = 0 on Ω and NOT, as one might expect, v = f (λ, v) Let me recall in passing that the limit of p-harmonic functions p u = 0 in Ω, u = g on Ω solves u = 0 in Ω, u = g on Ω in the sense of viscosity solutions. B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

10 Incidentally, the equation u = 0 was discovered by Gunnar Aronsson from Linköping as modeling constant slope in the shape of dry sandpiles. It is now called Aronsson s equation. An explicit C 1,α solution with α = 1/3 in 2d is u(x, y) = x 4/3 y 4/3. It was found by Aronsson and is not C 2. It is a viscosity solution. It describes the saddle that is formed when two sandpiles merge. Kawohl, Shagholian 2005 B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

11 It is instructive to look at the one-dim. case in min{ v λv, v} = 0 in Ω, v = 0 on Ω. For Ω = ( 1, 1) we have R(Ω) = 1, λ = 1 and v = 1 x as eigenfunction. For x (0, 1) any C 2 -testfunction φ touching v from above in x satisfies φ (x) φ(x) = 1 v(x) = x > 0 and φ (x) 2 φ (x) 0; and in x = 0 it might have φ (0) > 0, but φ (0) 1 = φ(0), so v is a viscosity subsolution. In a similar way one can look at C 2 -functions ψ touching v from below in x. For x = 0 there are no such functions, so v is also a viscosity supersolution. For p (1, ) the first eigenfunction is simple, but not for p = or p = 1. B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

12 The limit p 1 in the first Dirichlet eigenvalue leads to Cheeger sets, another geometric problem. Cheeger sets Ω C of Ω minimize perimeter / volume among all subsets of Ω, and after scaling the Dirichlet eigenfunctions u p converge to χ ΩC. NUMERICAL INVESTIGATION OF EIGENVALUES OF THE p-laplace OPERATOR 13 p = 1.1 p = 1.4 p = 2.0 p = 2.5 p = 8.0 For Ω convex, Figureis8. ΩThe c convex numerically andcomputed unique? first Yes, eigenfunction but short u 1 proof for thewanted. square. B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

13 What about higher eigenfunctions, at least in one dimension? These are well studied sin p and cos p -functions, satisfying identities like (sin p (z)) p p 1 + (cos q(z)) q q 1 = 1 where 1 p + 1 q = 1, and the n-th sin p -function on Ω = ( 1, 1) has n 1 zeroes in Ω. They have already been studied by Erik Lundberg in A reprint of his paper was preserved in the huge collection of reprints of Mittag-Leffler and is now kept in Lund. B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

14 What about a second eigenfunction in 2 dimensions? It can be characterized as a mountain pass going from u 1 to u 1, as shown by Cuesta, de Figueredo and Gossez. It changes sign, and so it has a nodal line. On a square and for p = 2 the nodal line can be horizontal, vertical or diagonal. On a disc it is known to be the diameter. On a square and for p = the diagonal provides larger inradius for nodal subdomains, and for p (2, ] the diagonal seems to be the preferred nodal pattern. Ω a square, p = 5, courtesy of J. Horák B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

15 On a square and for p = 1 the horizontal/resp. vertical provides the shortest (area-)halving length, and for p [1, 2) it is preferred. Ω a square, p = 1.1, courtesy of J. Horák B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33

16 On a disc and for p = one can think of nodal patterns like or, while for p (1, ) it cannot be a circle (Anoop, Drabek, Sasi 2016). Recently (in JDE 263 (2017) p ) Bobkov and Drabek showed that there can be higher eigenfunctions with radial-diameter nodal patterns like in the case p = 2 for general p (1, ), but that there can also be other nodal patterns for large enough p. B. Kawohl (Universität zu Köln) Eigenfunctions for versions of the p-laplacian Djursholm, May 17, / 33