Discrete Serrins Problem A. Carmona, A.M. Encinas and C. Ara uz - PowerPoint PPT Presentation

Discrete Serrin’s Problem A. Carmona, A.M. Encinas and C. Ara´ uz Dept. Matem` atica Aplicada III

Discrete Serrin’s Problem Introduction Motivation The original Serrin’s Problem R n , with smooth boundary δ (Ω) , if u is the unique Given Ω ⊂ I solution of − ∆( u ) = 1 on Ω u = 0 on δ (Ω) then ∂u ∂ n is constant iff Ω is a ball and u ( x ) = 1 2 n ( R 2 −| x | 2 ) ; that is, u is radial. CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Motivation The original Serrin’s Problem R n , with smooth boundary δ (Ω) , if u is the unique Given Ω ⊂ I solution of − ∆( u ) = 1 on Ω u = 0 on δ (Ω) then ∂u ∂ n is constant iff Ω is a ball and u ( x ) = 1 2 n ( R 2 −| x | 2 ) ; that is, u is radial. ◮ Moving planes ◮ Minimum principle and Green Identities CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Motivation Discrete Serrin’s Problem � � Given Γ = F ∪ δ ( F ) , c a network with boundary if u is the unique solution of L ( u ) = 1 on F u = 0 on δ ( F ) then if ∂u ∂ n is constant, what can we say about Γ and u ? CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Motivation Discrete Serrin’s Problem � � Given Γ = F ∪ δ ( F ) , c a network with boundary if u is the unique solution of L ( u ) = 1 on F u = 0 on δ ( F ) then if ∂u ∂ n is constant, what can we say about Γ and u ? ◮ Minimum principle and Green Identities CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Motivation Discrete Serrin’s Problem � � Given Γ = F ∪ δ ( F ) , c a network with boundary if u is the unique solution of L ( u ) = 1 on F u = 0 on δ ( F ) then if ∂u ∂ n is constant, what can we say about Γ and u ? ◮ Minimum principle and Green Identities ◮ Existence of equilibrium measure CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Notations Network Topology ◮ Network Γ = ( V, E, c ) CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Notations Network Topology ◮ Network Γ = ( V, E, c ) ◮ Given F ⊂ V consider the sets Ext ( F ) δ ( F ) D 1  D 2         ◦  D 3 F  r ( F )= max x ∈ F { d ( x, δ ( F ) }      D 4    CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Notations Operators ◮ Combinatorial Laplacian L : C ( V ) − → C ( V ) � � � � L ( u )( x )= c ( x, y ) u ( x ) − u ( y ) = k ( x ) u ( x ) − c ( x, y ) u ( y ) y ∈ V y ∈ V CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Notations Operators ◮ Combinatorial Laplacian L : C ( V ) − → C ( V ) � � � � L ( u )( x )= c ( x, y ) u ( x ) − u ( y ) = k ( x ) u ( x ) − c ( x, y ) u ( y ) y ∈ V y ∈ V ◮ Matrix version   k ( x 1 ) − c ( x 1 , x 2 ) · · · − c ( x 1 , x n )   − c ( x 1 , x 2 ) k ( x 2 ) · · · − c ( x 2 , x n )     = D − A . . .  ...  . . .  . . .    − c ( x 1 , x n ) − c ( x 2 , x n ) · · · k ( x n ) CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Notations Operators ◮ Combinatorial Laplacian L : C ( V ) − → C ( V ) � � � � L ( u )( x )= c ( x, y ) u ( x ) − u ( y ) = k ( x ) u ( x ) − c ( x, y ) u ( y ) y ∈ V y ∈ V ◮ Normal derivative: u ∈ C ( V ) and F connected proper set ∂u � � � ∂ n ( x ) = c ( x, y ) u ( x ) − u ( y ) , for any x ∈ δ ( F ) y ∈ F CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Notations Operators ◮ Combinatorial Laplacian L : C ( V ) − → C ( V ) � � � � L ( u )( x )= c ( x, y ) u ( x ) − u ( y ) = k ( x ) u ( x ) − c ( x, y ) u ( y ) y ∈ V y ∈ V ◮ Normal derivative: u ∈ C ( V ) and F connected proper set ∂u � � � ∂ n ( x ) = c ( x, y ) u ( x ) − u ( y ) , for any x ∈ δ ( F ) y ∈ F ∂u � � Gauss Theorem: L ( u )( x ) = − ∂ n ( x ) x ∈ F x ∈ δ ( F ) CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Basic Results Minimum principle ◮ A function u ∈ C ( V ) is called ⊲ Superharmonic if L ( u ) ≥ 0 CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Basic Results Minimum principle ◮ A function u ∈ C ( V ) is called ⊲ Superharmonic if L ( u ) ≥ 0 ⊲ Strictly Superharmonic if L ( u ) > 0 CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Introduction Basic Results Minimum principle ◮ A function u ∈ C ( V ) is called ⊲ Superharmonic if L ( u ) ≥ 0 ⊲ Strictly Superharmonic if L ( u ) > 0 If u ∈ C ( V ) is superharmonic on F , then x ∈ δ ( F ) { u ( x ) } ≤ min min x ∈ F { u ( x ) } ◮ The equality holds iff u = aχ ¯ F CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Generalized minimum principles If u ∈ C ( ¯ F ) is superharmonic on F , then for any i = 1 , . . . , r ( F ) − 1 ◮ x ∈ δ ( F ) { u ( x ) } ≤ min min x ∈ D i { u ( x ) } ≤ x ∈ D i +1 { u ( x ) } min CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Generalized minimum principles If u ∈ C ( ¯ F ) is superharmonic on F , then for any i = 1 , . . . , r ( F ) − 1 ◮ x ∈ δ ( F ) { u ( x ) } ≤ min min x ∈ D i { u ( x ) } ≤ x ∈ D i +1 { u ( x ) } min δ ( F ) 0 0 0 D 1 23 . 1 22 . 7 9 21 . 3 17  D 2 22 . 5        ◦   D 3 F 22 19 . 7       D 4   21 . 4  CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Generalized minimum principles If u ∈ C ( ¯ F ) is superharmonic on F , then for any i = 1 , . . . , r ( F ) − 1 ◮ x ∈ δ ( F ) { u ( x ) } ≤ min min x ∈ D i { u ( x ) } ≤ x ∈ D i +1 { u ( x ) } min If u ∈ C + ( F ) is a strictly superharmonic function on F , then for any x ∈ F there exists y ∈ ¯ F such that c ( x, y ) > 0 and ◮ u ( y ) < u ( x ) CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Generalized minimum principles If u ∈ C + ( F ) is a strictly superharmonic function on F , then for any x ∈ F there exists y ∈ ¯ F such that c ( x, y ) > 0 and ◮ u ( y ) < u ( x ) δ ( F ) 0 0 0 D 1 23 . 1 22 . 7 9 21 . 3  17 D 2 22 . 5        ◦   D 3 19 . 7 F 22      D 4    21 . 4  CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Level sets ◮ Given u ∈ C + ( F ) we denote 0 = u 0 < u 1 < · · · < u s ⊲ Level set U i = { x ∈ F : u ( x ) = u i } CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Level sets ◮ Given u ∈ C + ( F ) we denote 0 = u 0 < u 1 < · · · < u s ⊲ Level set U i = { x ∈ F : u ( x ) = u i } If u ∈ C + ( F ) is a strictly superharmonic function on F , then i ◮ � U 0 = D 0 and U i ⊂ D i , for any i = 1 , . . . , s j =1 CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Level sets ◮ Given u ∈ C + ( F ) we denote 0 = u 0 < u 1 < · · · < u s ⊲ Level set U i = { x ∈ F : u ( x ) = u i } If u ∈ C + ( F ) is a strictly superharmonic function on F , then i ◮ � U 0 = D 0 and U i ⊂ D i , for any i = 1 , . . . , s j =1 δ ( F ) 0 0 0 0 5 5 1 1 3 3 U 2 D 2 7 5 3 3 CJI 2013 (RSME), Sevilla 10 3

Discrete Serrin’s Problem Superharmonic functions Minimum principles Level sets ◮ Given u ∈ C + ( F ) we denote 0 = u 0 < u 1 < · · · < u s ⊲ Level set U i = { x ∈ F : u ( x ) = u i } If u ∈ C + ( F ) is a strictly superharmonic function on F , then i ◮ � U 0 = D 0 and U i ⊂ D i , for any i = 1 , . . . , s j =1 If u ∈ C + ( F ) is a strictly superharmonic function on F ◮ satisfying U j = D j for all j = 0 , . . . , i, then U i +1 ⊂ D i +1 CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Radial Functions ◮ An strictly superharmonic function u ∈ C + ( F ) is called ◮ radial if U i = D i , for any i = 0 , . . . , s CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Radial Functions ◮ An strictly superharmonic function u ∈ C + ( F ) is called ◮ radial if U i = D i , for any i = 0 , . . . , s = ⇒ s = r ( F ) CJI 2013 (RSME), Sevilla

Discrete Serrin’s Problem Superharmonic functions Minimum principles Radial Functions ◮ An strictly superharmonic function u ∈ C + ( F ) is called ◮ radial if U i = D i , for any i = 0 , . . . , s = ⇒ s = r ( F ) If u ∈ C + ( F ) is a radial function , then for any x ∈ D i ◮ � � � � L u ( x ) = k i +1 ( x ) u i − u i +1 + k i − 1 ( x ) u i − u i − 1 > 0 , CJI 2013 (RSME), Sevilla