Nonlocal, nonlinear, nonsmooth Juan Pablo Borthagaray Department of Mathematjcs, University of Maryland, College Park Fractjonal PDEs: Theory, Algorithms and Applicatjons ICERM – Brown University June 22, 2018
Pointwise limits as s : s u lim u s s u lim u s Fractjonal Laplacian in R n Let s ∈ (0 , 1) and u : R n → R be smooth enough (belongs to Schwartz class). Pseudodifferentjal operator: F (( − ∆) s u ) ( ξ ) = | ξ | 2 s F u ( ξ ) . Integral representatjon: ˆ u ( x ) − u ( y ) ( − ∆) s u ( x ) = C ( n , s ) p.v. | x − y | n +2 s dy , R n where C ( n , s ) = 2 2 s s Γ( s + n 2 ) π n /2 Γ(1 − s ) is a normalizatjon constant. Probabilistjc interpretatjon: related to random walks with jumps. Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Fractjonal Laplacian in R n Let s ∈ (0 , 1) and u : R n → R be smooth enough (belongs to Schwartz class). Pseudodifferentjal operator: F (( − ∆) s u ) ( ξ ) = | ξ | 2 s F u ( ξ ) . Integral representatjon: ˆ u ( x ) − u ( y ) ( − ∆) s u ( x ) = C ( n , s ) p.v. | x − y | n +2 s dy , R n where C ( n , s ) = 2 2 s s Γ( s + n 2 ) π n /2 Γ(1 − s ) is a normalizatjon constant. Probabilistjc interpretatjon: related to random walks with jumps. Pointwise limits as s → 0 , 1 : s → 0 ( − ∆) s u = u , lim s → 1 ( − ∆) s u = − ∆ u . lim Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Integral definitjon for Ω ⊂ R n Let Ω ⊂ R n be an open bounded set, and let f : Ω → R . Boundary value problem: � ( − ∆) s u = f in Ω , in Ω c . u = 0 Integral representatjon: u ( x ) − u ( y ) ˆ ( − ∆) s u ( x ) = C ( n , s ) p.v. | x − y | n +2 s dy = f ( x ) , x ∈ Ω . R n Boundary conditjons: imposed in Ω c = R n \ Ω in Ω c . u = 0 Probabilistjc interpretatjon: it is the same as over R n except that partjcles are killed upon reaching Ω c . Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Between the identjty and the Laplacian Solutjons to fractjonal obstacle problems on the square [ − 1 , 1] × [ − 1 , 1] , with f = 0 , various s , and obstacle � 1 � � �� � � 1 � � 1 �� � � � � � − 3 4 , 3 4 , − 1 � � � � 4 − � x − 2 − � x − χ ( x ) = max � , 0 + max � , 0 . 4 4 Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Some remarks There is not a unique way to define a “fractjonal Laplacian” over Ω (spectral, restricted, tempered, directjonal...). Numerical methods for the integral fractjonal Laplacian on bounded domains include ◮ Finite elements (on integral representatjon): D’Elia & Gunzburger (2013), Ainsworth & Glusa (2018). ◮ Finite differences: Huang & Oberman (2014), Duo, van Wyk & Zhang (2018). ◮ Walk-on-spheres method: Kyprianou, Osojnik & Shardlow (2017). ◮ Collocatjon methods: Zeng, Zhang & Karniadakis (2015), Acosta, B., Bruno & Maas (2018) ◮ Finite elements (using Dunford-Taylor representatjon): Bonito, Lei & Pasciak (2017). ◮ . . . (To the best of my knowledge) these methods have been implemented mainly for linear/semilinear problems. Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Goal & outline Design finiteelement methods for nonlocal(fractjonal)problems. Derive Sobolev regularity estjmates and perform a finite element analysis of these problems on bounded domains. (Linear) Dirichlet problem. ◮ Regularity of solutjons. ◮ Finite element discretjzatjons. ◮ Reduced regularity near ∂ Ω : graded meshes. Fractjonal obstacle problem. Fractjonal minimal surfaces. Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Functjon spaces Fractjonal Sobolev spaces in R n : � � H s ( R n ) = v ∈ L 2 ( R n ): | v | H s ( R n ) < ∞ with � u , w � := C ( n , s ) ( u ( x ) − u ( y ))( v ( x ) − v ( y )) ¨ dydx , | x − y | n +2 s 2 R n × R n � � 1 2 . 1 2 , � v � 2 L 2 ( R n ) + | v | 2 | v | H s ( R n ) := � v , v � � v � H s ( R n ) := H s ( R n ) Fractjonal Sobolev spaces in Ω : � � � H s (Ω) := v | Ω : v ∈ H s ( R n ) , supp ( v ) ⊂ Ω � v � � H s (Ω) := � v � H s ( R n ) . , � � ∗ � Dual space: H − s (Ω) = H s (Ω) . Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
(see next slide) H s H Galerkin orthogonality Lagrange interpolatjon quasi-interpolatjon Nonlocality The H s -seminorms are not additjve with respect to domain partjtjons. Functjons with disjoint supports may have a non-zero inner product: if u v on their supports, then C n s u x v y u v s dx dy x y n supp u supp v Singular integrals, integratjon on unbounded domains. How smooth are solutjons? Is there a lifuing property? Something old, something new... All the basic analysis tools we need have a fractjonal counterpart! ◮ Integratjon by parts formula ◮ Coercive bilinear form on a suitable space (Poincaré inequality) ◮ Finite elements = projectjon w.r.t. energy norm ◮ Interpolatjon estjmates Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Nonlocality The H s -seminorms are not additjve with respect to domain partjtjons. Functjons with disjoint supports may have a non-zero inner product: if u v on their supports, then C n s u x v y u v s dx dy x y n supp u supp v Singular integrals, integratjon on unbounded domains. How smooth are solutjons? Is there a lifuing property? Something old, something new... All the basic analysis tools we need have a fractjonal counterpart! ◮ Integratjon by parts formula (see next slide) � ◮ Coercive bilinear form on a suitable space (Poincaré inequality) H 1 0 (Ω) �→ � H s (Ω) � ◮ Finite elements = projectjon w.r.t. energy norm Galerkin orthogonality � ◮ Interpolatjon estjmates Lagrange interpolatjon �→ quasi-interpolatjon � Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Something old, something new... All the basic analysis tools we need have a fractjonal counterpart! ◮ Integratjon by parts formula (see next slide) � ◮ Coercive bilinear form on a suitable space (Poincaré inequality) H 1 0 (Ω) �→ � H s (Ω) � ◮ Finite elements = projectjon w.r.t. energy norm Galerkin orthogonality � ◮ Interpolatjon estjmates Lagrange interpolatjon �→ quasi-interpolatjon � Nonlocality ◮ The H s -seminorms are not additjve with respect to domain partjtjons. ◮ Functjons with disjoint supports may have a non-zero inner product: if u , v > 0 on their supports, then � u , v � = C ( n , s ) ¨ − 2 u ( x ) v ( y ) | x − y | n +2 s dx dy < 0 . 2 supp ( u ) × supp ( v ) ◮ Singular integrals, integratjon on unbounded domains. ◮ How smooth are solutjons? Is there a lifuing property? Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
c , it may return to any point Random walk interpretatjon: if the partjcle goes to x n s . y , with the probability of jumping from x to y being proportjonal to x y c into The functjon s u can be regarded as a nonlocal flux density on . Integratjon by parts (Dipierro, Ros-Oton & Valdinoci (2017)) ˆ ˆ v ( x )( − ∆) s u ( x ) dx + Ω c v ( x ) N s u ( x ) dx . � u , v � = Ω Here, � u , v � := C ( n , s ) ( u ( x ) − u ( y ))( v ( x ) − v ( y )) ¨ dx dy , | x − y | n +2 s 2 ( R n × R n ) \ (Ω c × Ω c ) and N s is a nonlocal derivatjve operator, ˆ u ( x ) − u ( y ) x ∈ Ω c . N s u ( x ) := C ( n , s ) | x − y | n +2 s dy , Ω Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Integratjon by parts (Dipierro, Ros-Oton & Valdinoci (2017)) ˆ ˆ v ( x )( − ∆) s u ( x ) dx + Ω c v ( x ) N s u ( x ) dx . � u , v � = Ω Here, � u , v � := C ( n , s ) ( u ( x ) − u ( y ))( v ( x ) − v ( y )) ¨ dx dy , | x − y | n +2 s 2 ( R n × R n ) \ (Ω c × Ω c ) and N s is a nonlocal derivatjve operator, ˆ u ( x ) − u ( y ) x ∈ Ω c . N s u ( x ) := C ( n , s ) | x − y | n +2 s dy , Ω Random walk interpretatjon: if the partjcle goes to x ∈ Ω c , it may return to any point y ∈ Ω , with the probability of jumping from x to y being proportjonal to | x − y | − n − 2 s . The functjon N s u can be regarded as a nonlocal flux density on Ω c into Ω . Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Variatjonal formulatjon: H s u v f v v s H s where stands for the duality pairing H . Poincaré inequality in H s : H s v L c n s v H s v n H s H s is an inner product in H s Therefore, the form , and we will write v H s v v . Existence, uniqueness, and stability follow from Lax-Milgram theorem. Dirichlet problem Given f ∈ H − s (Ω) , find u ∈ � H s (Ω) such that � ( − ∆) s u = f in Ω , in Ω c . u = 0 Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
Dirichlet problem Given f ∈ H − s (Ω) , find u ∈ � H s (Ω) such that � ( − ∆) s u = f in Ω , in Ω c . u = 0 Variatjonal formulatjon: ∀ v ∈ � H s (Ω) , � u , v � = ( f , v ) where ( · , · ) stands for the duality pairing H − s (Ω) × � H s (Ω) . Poincaré inequality in � H s (Ω) : ∀ v ∈ � H s (Ω) . � v � L 2 (Ω) ≤ c (Ω , n , s ) | v | H s ( R n ) Therefore, the form � · , · � : � H s (Ω) × � H s (Ω) is an inner product in � H s (Ω) , and we H s (Ω) = � v , v � 1/2 . will write � v � � Existence, uniqueness, and stability follow from Lax-Milgram theorem. Juan Pablo Borthagaray Nonlocal, nonlinear, nonsmooth
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