Prescribed Energy Solutions of some class of Semilinear Elliptic - PowerPoint PPT Presentation

Prescribed Energy Solutions of some class of Semilinear Elliptic Equations Piero Montecchiari Universit` a Politecnica delle Marche joint work with Francesca Alessio

De Giorgi Conjecture (’78): If − ∆ u = u − u 3 on R n , − 1 < u < 1, ∂ x n u > 0 and n ≤ 8 then u ( x ) = q ( a · x + b ) for an a ∈ R n and b ∈ R q = q − q 3 on R where q solves − ¨ Gibbons conjecture: If − ∆ u = u − u 3 on R n , and x n →±∞ u ( x ) = ± 1 uniformly w.r.t ( x 1 , . . . , x n − 1 ) ∈ R n − 1 lim then q = q − q 3 on R u ( x ) = q ( x n ) where q solves − ¨

De Giorgi Conjecture (’78): If − ∆ u = u − u 3 on R n , − 1 < u < 1, ∂ x n u > 0 and n ≤ 8 then u ( x ) = q ( a · x + b ) for an a ∈ R n and b ∈ R q = q − q 3 on R where q solves − ¨ Gibbons conjecture: If − ∆ u = u − u 3 on R n , and x n →±∞ u ( x ) = ± 1 uniformly w.r.t ( x 1 , . . . , x n − 1 ) ∈ R n − 1 lim then q = q − q 3 on R u ( x ) = q ( x n ) where q solves − ¨

Proof of Gibbons Conjecture Farina, Ricerche di Matematica (in honour of E. De Giorgi) ’98 Barlow, Bass, Gui, CPAM ’00 Berestycki, Hamel, Monneau, Duke ’00 Proof of De Giorgi Conjecture n = 2 Ghoussoub, Gui, Math. Ann. ’98 n = 3 Ambrosio, Cabr` e, JAMS ’00 n ≤ 8 Savin, Ann. of Math. ’09 (PhD Thesis ’03) (assuming lim x n →±∞ u = ± 1) n > 8 Del Pino, Kowalczyk, Wei, Preprint (C.R. Math. Acad. Paris ’08)

Proof of Gibbons Conjecture Farina, Ricerche di Matematica (in honour of E. De Giorgi) ’98 Barlow, Bass, Gui, CPAM ’00 Berestycki, Hamel, Monneau, Duke ’00 Proof of De Giorgi Conjecture n = 2 Ghoussoub, Gui, Math. Ann. ’98 n = 3 Ambrosio, Cabr` e, JAMS ’00 n ≤ 8 Savin, Ann. of Math. ’09 (PhD Thesis ’03) (assuming lim x n →±∞ u = ± 1) n > 8 Del Pino, Kowalczyk, Wei, Preprint (C.R. Math. Acad. Paris ’08)

Border examples De Giorgi setting Del Pino, Kowalczyk, Pacard, Wei, JFA ’10 Infinitely many non monotone multidimensional solutions Border examples Gibbons setting Systems of Allen Cahn equations Alama, Bronsard, Gui, CVPDE ’97 Schatzman, COCV ’02 Existence of one multidimensional solution Equations with potential depending on the x n variable Alessio, JeanJean, M. CVPDE ’00 Alessio, M., COCV ’05, ANS ’05, CVPDE ’07 Infinitely many (monotone) bidimensional solutions

Border examples De Giorgi setting Del Pino, Kowalczyk, Pacard, Wei, JFA ’10 Infinitely many non monotone multidimensional solutions Border examples Gibbons setting Systems of Allen Cahn equations Alama, Bronsard, Gui, CVPDE ’97 Schatzman, COCV ’02 Existence of one multidimensional solution Equations with potential depending on the x n variable Alessio, JeanJean, M. CVPDE ’00 Alessio, M., COCV ’05, ANS ’05, CVPDE ’07 Infinitely many (monotone) bidimensional solutions

Border examples De Giorgi setting Del Pino, Kowalczyk, Pacard, Wei, JFA ’10 Infinitely many non monotone multidimensional solutions Border examples Gibbons setting Systems of Allen Cahn equations Alama, Bronsard, Gui, CVPDE ’97 Schatzman, COCV ’02 Existence of one multidimensional solution Equations with potential depending on the x n variable Alessio, JeanJean, M. CVPDE ’00 Alessio, M., COCV ’05, ANS ’05, CVPDE ’07 Infinitely many (monotone) bidimensional solutions

Systems of Allen Cahn type equations Consider the system studied in [ABG97] ( x , y ) ∈ R 2 (S) − ∆ u ( x , y ) + ∇ W ( u ( x , y )) = 0 , where W ∈ C 2 ( R 2 , R ) is a double well potential: (W1) 0 = W ( ± 1 , 0) < W ( ξ ) for any ξ ∈ R 2 \{ ( ± 1 , 0) } , D 2 W ( ± 1 , 0) > 0, (W2) lim inf | ξ |→ + ∞ W ( ξ ) > 0, (W3) W ( − x , y ) = W ( x , y ) . Problem: find bidimensional solutions u satisfying x →±∞ u ( x , y ) = ( ± 1 , 0) lim unif. w.r.t. y ∈ R .

Systems of Allen Cahn type equations Consider the system studied in [ABG97] ( x , y ) ∈ R 2 (S) − ∆ u ( x , y ) + ∇ W ( u ( x , y )) = 0 , where W ∈ C 2 ( R 2 , R ) is a double well potential: (W1) 0 = W ( ± 1 , 0) < W ( ξ ) for any ξ ∈ R 2 \{ ( ± 1 , 0) } , D 2 W ( ± 1 , 0) > 0, (W2) lim inf | ξ |→ + ∞ W ( ξ ) > 0, (W3) W ( − x , y ) = W ( x , y ) . Problem: find bidimensional solutions u satisfying x →±∞ u ( x , y ) = ( ± 1 , 0) lim unif. w.r.t. y ∈ R .

Systems of Allen Cahn type equations Consider the system studied in [ABG97] ( x , y ) ∈ R 2 (S) − ∆ u ( x , y ) + ∇ W ( u ( x , y )) = 0 , where W ∈ C 2 ( R 2 , R ) is a double well potential: (W1) 0 = W ( ± 1 , 0) < W ( ξ ) for any ξ ∈ R 2 \{ ( ± 1 , 0) } , D 2 W ( ± 1 , 0) > 0, (W2) lim inf | ξ |→ + ∞ W ( ξ ) > 0, (W3) W ( − x , y ) = W ( x , y ) . Problem: find bidimensional solutions u satisfying x →±∞ u ( x , y ) = ( ± 1 , 0) lim unif. w.r.t. y ∈ R .

Associated ODE System: heteroclinic solutions � − ¨ q ( x ) + ∇ W ( q ( x )) = 0 , x ∈ R q ( ±∞ ) = ( ± 1 , 0) . We look for the minima of the Action � + ∞ 1 q | 2 + W ( q ) dx V ( q ) = 2 | ˙ −∞ over the space Γ = { q − ψ ∈ H 1 ( R ) 2 / q 1 ( x ) = − q 1 ( − x ) } ψ fixed such that ψ ( x ) = (1 , 0) for x > 1 and ψ ( x ) = ( − 1 , 0) for x < − 1.

Associated ODE System: heteroclinic solutions � − ¨ q ( x ) + ∇ W ( q ( x )) = 0 , x ∈ R q ( ±∞ ) = ( ± 1 , 0) . We look for the minima of the Action � + ∞ 1 q | 2 + W ( q ) dx V ( q ) = 2 | ˙ −∞ over the space Γ = { q − ψ ∈ H 1 ( R ) 2 / q 1 ( x ) = − q 1 ( − x ) } ψ fixed such that ψ ( x ) = (1 , 0) for x > 1 and ψ ( x ) = ( − 1 , 0) for x < − 1.

Setting c = inf Γ V ( q ) then K = { q ∈ Γ / V ( q ) = c } � = ∅ . Discreteness Assumption ( H c ): K = K − ∪ K + with dist L 2 ( K − , K + ) > 0.

Setting c = inf Γ V ( q ) then K = { q ∈ Γ / V ( q ) = c } � = ∅ . Discreteness Assumption ( H c ): K = K − ∪ K + with dist L 2 ( K − , K + ) > 0.

Setting c = inf Γ V ( q ) then K = { q ∈ Γ / V ( q ) = c } � = ∅ . Discreteness Assumption ( H c ): K = K − ∪ K + with dist L 2 ( K − , K + ) > 0.

Setting c = inf Γ V ( q ) then K = { q ∈ Γ / V ( q ) = c } � = ∅ . Discreteness Assumption ( H c ): K = K − ∪ K + with dist L 2 ( K − , K + ) > 0.

Looking for bidimensional solutions. We look for bidimensional solutions prescribing different asymptots as y → ±∞ : dist L 2 ( u ( · , y ) , K ± ) → 0 as y → ±∞ .

Looking for bidimensional solutions. We look for bidimensional solutions prescribing different asymptots as y → ±∞ : dist L 2 ( u ( · , y ) , K ± ) → 0 as y → ±∞ .

Looking for bidimensional solutions. We look for bidimensional solutions prescribing different asymptots as y → ±∞ : dist L 2 ( u ( · , y ) , K ± ) → 0 as y → ±∞ .

Theorem. If (W1)- (W3) and ( H c ) are satisfied then there exists a bidimensional solution. Sketch of the Proof: Variational settings: We choose the variational space prescribing the right limits at infinity: loc ( R 2 ) 2 | u ( · , y ) ∈ Γ for a.e. y ∈ R } M = { u ∈ H 1 X = { u ∈ M | lim inf y →±∞ dist L 2 ( u ( · , y ) , K ± ) = 0 } Remark: The Euler Lagrange functional if always infinite on X : � � 2 | ∂ y u | 2 + 1 2 | ∂ x u | 2 + W ( u ) dx dy 1 R R �� � � � 2 | ∂ y u | 2 dx + 2 | ∂ x u | 2 + W ( u ) dx 1 1 = dy R R R � 1 2 � ∂ y u ( · , y ) � 2 L 2 ( R ) 2 + V ( u ( · , y )) dy = + ∞ ∀ u ∈ X = R

Theorem. If (W1)- (W3) and ( H c ) are satisfied then there exists a bidimensional solution. Sketch of the Proof: Variational settings: We choose the variational space prescribing the right limits at infinity: loc ( R 2 ) 2 | u ( · , y ) ∈ Γ for a.e. y ∈ R } M = { u ∈ H 1 X = { u ∈ M | lim inf y →±∞ dist L 2 ( u ( · , y ) , K ± ) = 0 } Remark: The Euler Lagrange functional if always infinite on X : � � 2 | ∂ y u | 2 + 1 2 | ∂ x u | 2 + W ( u ) dx dy 1 R R �� � � � 2 | ∂ y u | 2 dx + 2 | ∂ x u | 2 + W ( u ) dx 1 1 = dy R R R � 1 2 � ∂ y u ( · , y ) � 2 L 2 ( R ) 2 + V ( u ( · , y )) dy = + ∞ ∀ u ∈ X = R

Theorem. If (W1)- (W3) and ( H c ) are satisfied then there exists a bidimensional solution. Sketch of the Proof: Variational settings: We choose the variational space prescribing the right limits at infinity: loc ( R 2 ) 2 | u ( · , y ) ∈ Γ for a.e. y ∈ R } M = { u ∈ H 1 X = { u ∈ M | lim inf y →±∞ dist L 2 ( u ( · , y ) , K ± ) = 0 } Remark: The Euler Lagrange functional if always infinite on X : � � 2 | ∂ y u | 2 + 1 2 | ∂ x u | 2 + W ( u ) dx dy 1 R R �� � � � 2 | ∂ y u | 2 dx + 2 | ∂ x u | 2 + W ( u ) dx 1 1 = dy R R R � 1 2 � ∂ y u ( · , y ) � 2 L 2 ( R ) 2 + V ( u ( · , y )) dy = + ∞ ∀ u ∈ X = R