Invertible Harmonic Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction Invertible Harmonic Mappings in the Plane Higher Dimensions Elliptic Operators Giovanni Alessandrini 1 Vincenzo Nesi 2 Elliptic Systems Non-convex 1 Target Università di Trieste The counter- 2 Università La Sapienza di Roma example Open issues The 4th Symposium on Analysis & PDEs, Purdue 2009 End
Invertible Harmonic The Basic Question Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction Let B ⊂ R 2 be the unit disk. Let D ⊂ R 2 be a Jordan Higher Dimensions domain. Elliptic Operators Given a homeomorphism Φ : ∂ B �→ ∂ D , consider the solution U = ( u 1 , u 2 ) : B �→ R 2 to the following Dirichlet Elliptic Systems problem Non-convex � ∆ U = 0 , Target in B , The counter- U = Φ , on ∂ B . example Open issues Under which conditions on Φ do we have that End U is a homeomorphism of B �→ D ?
Invertible Harmonic The Classical Results Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction Φ : ∂ B �→ ∂ D , Higher � ∆ U = 0 , Dimensions in B , Elliptic U = Φ , on ∂ B . Operators Elliptic Systems Theorem ( H. Kneser ’26) Non-convex Target If D is convex, then U is a homeomorphism of B onto D. The counter- Posed as a problem by Radó (’26), rediscovered by example Choquet (’45). Open issues End Theorem (H. Lewy ’36) If U : B �→ R 2 is a harmonic homeomorphism, then it is a diffeomorphism.
Invertible Harmonic The Classical Results Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction Φ : ∂ B �→ ∂ D , Higher � ∆ U = 0 , Dimensions in B , Elliptic U = Φ , on ∂ B . Operators Elliptic Systems Theorem ( H. Kneser ’26) Non-convex Target If D is convex, then U is a homeomorphism of B onto D. The counter- Posed as a problem by Radó (’26), rediscovered by example Choquet (’45). Open issues End Theorem (H. Lewy ’36) If U : B �→ R 2 is a harmonic homeomorphism, then it is a diffeomorphism.
Invertible Harmonic The Classical Results Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction Φ : ∂ B �→ ∂ D , Higher � ∆ U = 0 , Dimensions in B , Elliptic U = Φ , on ∂ B . Operators Elliptic Systems Theorem ( H. Kneser ’26) Non-convex Target If D is convex, then U is a homeomorphism of B onto D. The counter- Posed as a problem by Radó (’26), rediscovered by example Choquet (’45). Open issues End Theorem (H. Lewy ’36) If U : B �→ R 2 is a harmonic homeomorphism, then it is a diffeomorphism.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Natural questions. Mappings Giovanni Alessandrini, Vincenzo Nesi • What happens in higher dimensions? Introduction Higher • Can we replace ∆ with other elliptic operators? Dimensions Elliptic • Can we replace the diagonal ∆ system with other Operators elliptic systems? Elliptic Systems • Can we dispense with the convexity of the target D ? Non-convex Target Motivations The counter- example • Minimal surfaces. Open issues • Inverse problems. End • Homogenization. • Variational grid generation.
Invertible Harmonic Higher Dimensions. Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction • Wood (’74): There exists a harmonic homeomorphism U : R 3 �→ R 3 such that det DU ( 0 ) = 0. Higher Dimensions • Melas (’93): There exists a harmonic homeomorphism Elliptic Operators U : B �→ B , B ⊂ R 3 unit ball, such that det DU ( 0 ) = 0. Elliptic Systems • Laugesen (’96): ∀ ε > 0 ∃ Φ : ∂ B �→ ∂ B Non-convex homeomorphism, such that | Φ( x ) − x | < ε, ∀ x ∈ ∂ B Target and the solution U to The counter- example � ∆ U = 0 , Open issues in B , End U = Φ , on ∂ B . is not one-to-one.
Invertible Harmonic Higher Dimensions. Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction • Wood (’74): There exists a harmonic homeomorphism U : R 3 �→ R 3 such that det DU ( 0 ) = 0. Higher Dimensions • Melas (’93): There exists a harmonic homeomorphism Elliptic Operators U : B �→ B , B ⊂ R 3 unit ball, such that det DU ( 0 ) = 0. Elliptic Systems • Laugesen (’96): ∀ ε > 0 ∃ Φ : ∂ B �→ ∂ B Non-convex homeomorphism, such that | Φ( x ) − x | < ε, ∀ x ∈ ∂ B Target and the solution U to The counter- example � ∆ U = 0 , Open issues in B , End U = Φ , on ∂ B . is not one-to-one.
Invertible Harmonic Higher Dimensions. Mappings Giovanni Alessandrini, Vincenzo Nesi Introduction • Wood (’74): There exists a harmonic homeomorphism U : R 3 �→ R 3 such that det DU ( 0 ) = 0. Higher Dimensions • Melas (’93): There exists a harmonic homeomorphism Elliptic Operators U : B �→ B , B ⊂ R 3 unit ball, such that det DU ( 0 ) = 0. Elliptic Systems • Laugesen (’96): ∀ ε > 0 ∃ Φ : ∂ B �→ ∂ B Non-convex homeomorphism, such that | Φ( x ) − x | < ε, ∀ x ∈ ∂ B Target and the solution U to The counter- example � ∆ U = 0 , Open issues in B , End U = Φ , on ∂ B . is not one-to-one.
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