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State-of-the-art Proof Generalization to Heisenberg groups O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS Annalisa Massaccesi Warwick, 13 - 07 - 2017 Joint works with Don Vittone Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV

  1. State-of-the-art Proof Generalization to Heisenberg groups O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS Annalisa Massaccesi Warwick, 13 - 07 - 2017 Joint works with Don Vittone Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  2. State-of-the-art Proof Generalization to Heisenberg groups State-of-the-art 1 Proof 2 Generalization to Heisenberg groups 3 Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  3. State-of-the-art Proof Generalization to Heisenberg groups Section 1 State-of-the-art Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  4. State-of-the-art Proof Generalization to Heisenberg groups T HE R ANK -O NE T HEOREM Definition. An integrable function u : Ω ⊂ R n → R m has (locally) bounded variation if Du = ( D 1 u , . . . , D n u ) is a Radon measure. We decompose Du = D a u + D s u = ∇ u L n + σ u | D s u | Theorem (Alberti, 1993). If u ∈ BV (Ω; R m ) , then rank ( σ u ( x )) = 1 for | D s u | -a.e. x ∈ Ω . Applications to (vector) variational problems and systems of PDEs. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  5. State-of-the-art Proof Generalization to Heisenberg groups T HE R ANK -O NE T HEOREM Definition. An integrable function u : Ω ⊂ R n → R m has (locally) bounded variation if Du = ( D 1 u , . . . , D n u ) is a Radon measure. We decompose Du = D a u + D s u = ∇ u L n + σ u | D s u | Theorem (Alberti, 1993). If u ∈ BV (Ω; R m ) , then rank ( σ u ( x )) = 1 for | D s u | -a.e. x ∈ Ω . Applications to (vector) variational problems and systems of PDEs. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  6. State-of-the-art Proof Generalization to Heisenberg groups T HE R ANK -O NE T HEOREM Definition. An integrable function u : Ω ⊂ R n → R m has (locally) bounded variation if Du = ( D 1 u , . . . , D n u ) is a Radon measure. We decompose Du = D a u + D s u = ∇ u L n + σ u | D s u | Theorem (Alberti, 1993). If u ∈ BV (Ω; R m ) , then rank ( σ u ( x )) = 1 for | D s u | -a.e. x ∈ Ω . Applications to (vector) variational problems and systems of PDEs. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  7. State-of-the-art Proof Generalization to Heisenberg groups L ITERATURE AND REMARKS The rank-one property for BV functions was (1988) conjectured by E. De Giorgi and L. Ambrosio; (1993) proved by G. Alberti (see also Alberti-Csörnyei-Preiss); (2016) proved G. De Philippis and F. Rindler (consequence of a more general statement on A -free measures); (2016) proved by A. M. and D. Vittone. Remark. For SBV functions, i.e., D s u = ( u + − u − ) ν R H n − 1 � R with R rectifiable set, the theorem is way easier. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  8. State-of-the-art Proof Generalization to Heisenberg groups L ITERATURE AND REMARKS The rank-one property for BV functions was (1988) conjectured by E. De Giorgi and L. Ambrosio; (1993) proved by G. Alberti (see also Alberti-Csörnyei-Preiss); (2016) proved G. De Philippis and F. Rindler (consequence of a more general statement on A -free measures); (2016) proved by A. M. and D. Vittone. Remark. For SBV functions, i.e., D s u = ( u + − u − ) ν R H n − 1 � R with R rectifiable set, the theorem is way easier. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  9. State-of-the-art Proof Generalization to Heisenberg groups Section 2 Proof Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  10. State-of-the-art Proof Generalization to Heisenberg groups U SEFUL FACTS u : Ω → R has bounded variation if and only if the subgraph E u := { ( x , t ) ∈ Ω × R : t < u ( x ) } has finite perimeter in Ω × R . This is equivalent to χ E u ∈ BV (Ω × R ) . If u ∈ BV (Ω) , then D χ E u is rectifiable, more precisely D χ E u = ν E u H n � ∂ ∗ E u , where ∂ ∗ E u is the (rectifiable) reduced boundary and ν E u is the inner normal. See Miranda - several papers in the 60s; Giaquinta, Modica, Souˇ cek - Cartesian currents in the calculus of variations; Federer - Geometric Measure Theory. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  11. State-of-the-art Proof Generalization to Heisenberg groups U SEFUL FACTS u : Ω → R has bounded variation if and only if the subgraph E u := { ( x , t ) ∈ Ω × R : t < u ( x ) } has finite perimeter in Ω × R . This is equivalent to χ E u ∈ BV (Ω × R ) . If u ∈ BV (Ω) , then D χ E u is rectifiable, more precisely D χ E u = ν E u H n � ∂ ∗ E u , where ∂ ∗ E u is the (rectifiable) reduced boundary and ν E u is the inner normal. See Miranda - several papers in the 60s; Giaquinta, Modica, Souˇ cek - Cartesian currents in the calculus of variations; Federer - Geometric Measure Theory. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  12. State-of-the-art Proof Generalization to Heisenberg groups U SEFUL FACTS u : Ω → R has bounded variation if and only if the subgraph E u := { ( x , t ) ∈ Ω × R : t < u ( x ) } has finite perimeter in Ω × R . This is equivalent to χ E u ∈ BV (Ω × R ) . If u ∈ BV (Ω) , then D χ E u is rectifiable, more precisely D χ E u = ν E u H n � ∂ ∗ E u , where ∂ ∗ E u is the (rectifiable) reduced boundary and ν E u is the inner normal. See Miranda - several papers in the 60s; Giaquinta, Modica, Souˇ cek - Cartesian currents in the calculus of variations; Federer - Geometric Measure Theory. Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  13. State-of-the-art Proof Generalization to Heisenberg groups U SEFUL REMARK in Heisenberg Remark. Define V := { p ∈ ∂ ∗ E u : ( ν E u ( p )) n + 1 = 0 } . It is possible to see that H n -a.e. ( x , t ) ∈ V ν E u ( x , t ) = σ u ( x ) For instance: u ( x ) � x ∈ [ 0 , 1 ] Cantor staircase u ( x ) = π x ∈ ] 1 , 2 ] 0 E u ν E u log 2 x 3 − δ 1 = π ♯ ( ν E u H 1 � V ) . and Du = H log 3 � C 1 0 1 Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  14. State-of-the-art Proof Generalization to Heisenberg groups U SEFUL REMARK in Heisenberg Remark. Define V := { p ∈ ∂ ∗ E u : ( ν E u ( p )) n + 1 = 0 } . It is possible to see that ( D s u , 0 ) = π ♯ ( ν E u H n � V ) . For instance: u ( x ) � Cantor staircase x ∈ [ 0 , 1 ] π u ( x ) = 0 x ∈ ] 1 , 2 ] E u ν E u log 2 x 3 − δ 1 = π ♯ ( ν E u H 1 � V ) . and Du = H log 3 � C 1 0 1 Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  15. State-of-the-art Proof Generalization to Heisenberg groups P ROOF OF THE R ANK -O NE T HM . W.l.o.g. m = 2 and u = ( u 1 , u 2 ) . The polar vector σ u is multiple of ( σ 1 , σ 2 ) . If σ i ( x ) = 0 for some i , there is nothing to prove. Otherwise x = π ( p 1 ) = π ( p 2 ) for some p i ∈ V i and � ν E ui H n � V i � σ i | D s u i | = π ♯ i = 1 , 2 j = 1 Σ j i , where Σ j with V i ⊂ N i ∪ � ∞ i are surfaces.Then ( σ 1 ( x ) , 0 ) = ν E u 1 ( p 1 ) = ± ν Σ j 1 1 ( p 1 ) ( σ 2 ( x ) , 0 ) = ν E u 2 ( p 2 ) = ± ν Σ j 2 2 ( p 2 ) Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  16. State-of-the-art Proof Generalization to Heisenberg groups P ROOF OF THE R ANK -O NE T HM . W.l.o.g. m = 2 and u = ( u 1 , u 2 ) . The polar vector σ u is multiple of ( σ 1 , σ 2 ) . If σ i ( x ) = 0 for some i , there is nothing to prove. Otherwise x = π ( p 1 ) = π ( p 2 ) for some p i ∈ V i and � ν E ui H n � V i � σ i | D s u i | = π ♯ i = 1 , 2 j = 1 Σ j i , where Σ j with V i ⊂ N i ∪ � ∞ i are surfaces.Then ( σ 1 ( x ) , 0 ) = ν E u 1 ( p 1 ) = ± ν Σ j 1 1 ( p 1 ) ( σ 2 ( x ) , 0 ) = ν E u 2 ( p 2 ) = ± ν Σ j 2 2 ( p 2 ) Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  17. State-of-the-art Proof Generalization to Heisenberg groups P ROOF OF THE R ANK -O NE T HM . W.l.o.g. m = 2 and u = ( u 1 , u 2 ) . The polar vector σ u is multiple of ( σ 1 , σ 2 ) . If σ i ( x ) = 0 for some i , there is nothing to prove. Otherwise x = π ( p 1 ) = π ( p 2 ) for some p i ∈ V i and � ν E ui H n � V i � σ i | D s u i | = π ♯ i = 1 , 2 j = 1 Σ j i , where Σ j with V i ⊂ N i ∪ � ∞ i are surfaces.Then ( σ 1 ( x ) , 0 ) = ν E u 1 ( p 1 ) = ± ν Σ j 1 1 ( p 1 ) ( σ 2 ( x ) , 0 ) = ν E u 2 ( p 2 ) = ± ν Σ j 2 2 ( p 2 ) Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

  18. State-of-the-art Proof Generalization to Heisenberg groups P ROOF OF THE R ANK -O NE T HM . W.l.o.g. m = 2 and u = ( u 1 , u 2 ) . The polar vector σ u is multiple of ( σ 1 , σ 2 ) . If σ i ( x ) = 0 for some i , there is nothing to prove. Otherwise x = π ( p 1 ) = π ( p 2 ) for some p i ∈ V i and ν E ui H n � V i � � σ i | D s u i | = π ♯ i = 1 , 2 j = 1 Σ j i , where Σ j with V i ⊂ N i ∪ � ∞ i are surfaces.Then ( σ 1 ( x ) , 0 ) = ν E u 1 ( p 1 ) = ± ν Σ j 1 1 ( p 1 ) = − H n -a.e. by the following lemma ← ( σ 2 ( x ) , 0 ) = ν E u 2 ( p 2 ) = ± ν Σ j 2 2 ( p 2 ) Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS

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