The Dominative p -Laplacian Karl K. Brustad Aalto University KTH - PowerPoint PPT Presentation

The Dominative p -Laplacian Karl K. Brustad Aalto University KTH 26.-28. August 2019

The Dominative The p-Laplace Equation p -Laplacian Karl K. Brustad |∇ u | p − 2 ∇ u � � ∆ p u := div = 0 , 2 ≤ p < ∞ . Introduction The discovery Variational integral: by Crandall- Zhang ✂ |∇ u | p d x . An explanation of the superposition principle ∆ u = div ( ∇ u ) = 0 , p = 2 . Superposition of p -harmonic Infinity-Laplacian: functions Further ∆ ∞ u := ∇ u H u ∇ u T . applications of sublinear operators: Superposition Nonlinear when p > 2: in the doubly nonlinear diffusion ∆ p [ u + v ] � = ∆ p u + ∆ p v . equation

The Dominative The fundamental solution p -Laplacian Karl K. Brustad  p − n − p − 1 p − 1 , p − n | x | p � = n , Introduction    The discovery w n , p ( x ) := − ln | x | , p = n , by Crandall- Zhang   −| x | , p = ∞ or n = 1 .  An explanation of the superposition principle Superposition of p -harmonic functions Further applications of sublinear operators: Superposition in the doubly nonlinear diffusion equation Satisfies ∆ p w n , p ( x ) = 0 for x � = 0 in R n .

The Dominative Ellipticity & Viscosity p -Laplacian Karl K. Brustad Definition Introduction A function u : Ω → ( −∞ , ∞ ] is a (viscosity) supersolution to The discovery the equation D w = 0 in Ω if by Crandall- Zhang • u is lower semicontinuous. An explanation of • u is finite on a dense subset of Ω. the superposition • whenever φ ∈ C 2 touches u from below at some point principle x 0 ∈ Ω, then Superposition of p -harmonic D φ ( x 0 ) ≤ 0 . functions Further applications of sublinear Definition operators: Superposition An operator D u = F ( x , u , ∇ u , H u ) is (degenerate) elliptic if in the doubly nonlinear diffusion X ≤ Y ⇒ F ( x , s , q , X ) ≤ F ( x , s , q , Y ) equation for all ( x , s , q ) ∈ Ω × R × R n .

The Dominative Superposition principle for the p -Laplacian Karl K. fundamental solutions Brustad N Introduction � c i ≥ 0 , y i ∈ R n . V ( x ) := c i w n , p ( x − y i ) , The discovery by Crandall- i =1 Zhang An explanation of the superposition principle Superposition of p -harmonic functions Further applications of sublinear operators: Superposition in the doubly nonlinear diffusion equation Theorem (Crandall/Zhang (2003)) V is p-superharmonic in R n .

The Dominative Superposition principle for the p -Laplacian Karl K. fundamental solutions Brustad N Introduction � c i ≥ 0 , y i ∈ R n . V ( x ) := c i w n , p ( x − y i ) , The discovery by Crandall- i =1 Zhang Theorem (Crandall/Zhang (2003)) An explanation of the ∆ p V ≤ 0 . superposition principle Superposition Theorem (Lindqvist/Manfredi (2008)) of p -harmonic functions � ✂ � ∆ p w n , p ( x − y ) ρ ( y ) d y ≤ 0 , ρ ≥ 0 . Further applications of sublinear operators: Superposition in the doubly nonlinear N c i sin 2 θ i ( x ) diffusion ∆ p V ( x ) = − ( p − 2)( n + p − 2) � |∇ V ( x ) | p − 2 , equation p − 1 n + p − 2 | x − y i | p − 1 i =1 θ i ( x ) angle between x − y i and ∇ V ( x ).

The Dominative Preliminaries. p -Laplacian Karl K. Brustad The Hessian matrix: � ∂ 2 u Introduction � n H u := The discovery . by Crandall- ∂ x i ∂ x j i , j =1 Zhang An n real eigenvalues: explanation of the superposition principle λ 1 ≤ · · · ≤ λ n , λ i = λ i ( H u ) . Superposition of p -harmonic Bounds on the Rayleigh quotient: functions Further applications of λ 1 ≤ ξ T H u ξ sublinear 0 � = ξ ∈ R n . ≤ λ n , operators: | ξ | 2 Superposition in the doubly nonlinear diffusion equation

The Dominative The Dominative p -Laplacian p -Laplacian Karl K. Brustad Introduction ∆ p u = div( |∇ u | p − 2 ∇ u ) The discovery by Crandall- ( p − 2) ∇ u H u ∇ u T � � Zhang = |∇ u | p − 2 + ∆ u . |∇ u | 2 An explanation of the superposition principle D p u := ( p − 2) λ n ( H u ) + ∆ u , 2 ≤ p < ∞ . Superposition of p -harmonic Domination: functions Further ∆ p u ≤ |∇ u | p − 2 D p u ≤ 0 . applications of D p u ≤ 0 ⇒ sublinear operators: Superposition in the doubly nonlinear diffusion equation

The Dominative The Dominative p -Laplacian p -Laplacian Karl K. Brustad Sublinearity: Introduction D p [ u + v ] ≤ D p u + D p v , The discovery by Crandall- D p [ cu ] = c D p u , for c ≥ 0. Zhang An explanation of the A p : R n → S ( n ) , A p ( ξ ) := ( p − 2) ξξ T + I , superposition principle Superposition of p -harmonic functions D p u = max | ξ | =1 tr( A p ( ξ ) H u ) , Further applications of sublinear operators: D p u ≤ 0 , D p v ≤ 0 ⇒ D p [ u + v ] ≤ 0 . Superposition in the doubly nonlinear diffusion equation

The Dominative The Dominative p -Laplacian p -Laplacian Karl K. Brustad Equivalence: The gradient ∇ w n , p is an eigenvector to the Hessian H w n , p Introduction corresponding to the largest eigenvalue λ n ( H w n , p ). The discovery by Crandall- Zhang ∇ w n , p H w n , p ∇ w T An n , p = λ n ( H w n , p ) . explanation of |∇ w n , p | 2 the superposition principle Superposition D p w n , p = 0 . of p -harmonic functions Further applications of sublinear operators: Superposition in the doubly nonlinear diffusion equation

The Dominative An explanation of the p -Laplacian Karl K. superposition principle Brustad Introduction The discovery by Crandall- ∆ p w n , p = 0 and D p w n , p = 0 , (Equivalence) Zhang An D p u ≤ 0 , D p v ≤ 0 ⇒ D p [ u + v ] ≤ 0 , (Sublinearity) explanation of the D p w ≤ 0 ⇒ ∆ p w ≤ 0 . (Domination) superposition principle Superposition We immediately get of p -harmonic functions � N Further � applications of � ∆ p c i w n , p ( x − y i ) ≤ 0 . sublinear operators: i =1 Superposition in the doubly nonlinear diffusion equation

The Dominative Taking it further p -Laplacian Karl K. Brustad New questions. When is a generic sum � N i =1 u i p -superharmonic? Introduction Sufficient conditions: The discovery by Crandall- Zhang � N � An � D p u i ≤ 0 ⇒ ∆ p u i ≤ 0 . explanation of the i =1 superposition principle Superposition Necessary conditions?: of p -harmonic u ∈ C 2 is addable if functions Further 1 ∆ p [ u + l ] ≤ 0 for every linear function l ( x ) = a T x . applications of sublinear operators: 2 ∆ p [ u + cw n , p ◦ T ] ≤ 0 for every c ≥ 0 and every Superposition in the doubly translation T ( x ) = x − x 0 . nonlinear diffusion 3 ∆ p [ u + u ◦ T ] ≤ 0 for every isometry T . equation 1-3 is equivalent. 1-3 is equivalent to u being dominative p -superharmonic.

The Dominative Cylindrical functions p -Laplacian Karl K. Brustad Next question: What are the addable p - harmonic functions? • Addable = dominative p -superharmonic. Introduction The discovery • 0 = ∆ p u ≤ |∇ u | p − 2 D p u ≤ 0. by Crandall- Zhang The double equation: An explanation of the D p u = 0 = ∆ p u . superposition principle Superposition of p -harmonic functions A function f is radial if there exists a one-variable function F Further so that applications of sublinear f ( x ) = F ( | x | ) . operators: Superposition in the doubly nonlinear diffusion equation

The Dominative Cylindrical functions p -Laplacian Karl K. Brustad Definition A function f in R n is cylindrical (or k -cylindrical) if there exists Introduction The discovery a one-variable function F , an integer 1 ≤ k ≤ n , and an n × k by Crandall- Zhang matrix Q with orthonormal columns, i.e. Q T Q = I k , so that An explanation of � � the | Q T ( x − x 0 ) | f ( x ) = F superposition principle for some x 0 ∈ R n . We say that a function w in R n is a Superposition of p -harmonic functions cylindrical fundamental solution (to the p -Laplace Equation) Further if w is in the form applications of sublinear operators: � � Q T ( x − x 0 ) w ( x ) = C 1 w k , p + C 2 , C 1 ≥ 0 , Superposition in the doubly nonlinear diffusion for some k , Q and x 0 as above. equation D p w = 0 = ∆ p w .

The Dominative Cylindrical functions p -Laplacian Karl K. Brustad Theorem Introduction Let 2 < p < ∞ and let u ∈ C 2 (Ω) . If The discovery by Crandall- Zhang ∆ p u = 0 = D p u in Ω , An explanation of then u is locally a cylindrical fundamental solution. the superposition principle The addable p -harmonic functions are exactly the cylindrical Superposition fundamental solutions. of p -harmonic functions Further applications of sublinear operators: Superposition in the doubly nonlinear diffusion equation