On the Optimal Shape for the Heat Insulation Energy Problem Qinfeng - PowerPoint PPT Presentation

Introduction Existence of Optimal Shapes Stability Conclusion On the Optimal Shape for the Heat Insulation Energy Problem Qinfeng Li (Joint work with Hengrong Du and Prof. Changyou Wang) Department of Mathematics Purdue University PDE Seminar, 2017 1 / 23

Introduction Existence of Optimal Shapes Background :Optimization Problems in Thermal Insulation Stability The Asymptotic Problem Conclusion Motivation Σ ǫ := { σ + t ν ( σ ) | σ ∈ ∂ Ω , 0 ≤ t ≤ ǫ h ( σ ) } is the layer. Ω thermally conducting body, u temperature function, f ∈ L 2 (Ω) heat source, and h insulation material function of total amount m > 0, that is, h ∈ H m , where � H m = { h ≥ 0 : h ( σ ) d σ = m } . ∂ Ω 2 / 23

Introduction Existence of Optimal Shapes Background :Optimization Problems in Thermal Insulation Stability The Asymptotic Problem Conclusion Thermal Insulation Problem We want to design the optimal shape of Ω and find the optimal distribution of insulation material surrounding Ω . We assume the layer is very thin compared to the size of Ω , hence the study of the thermal insulation problem is to mathematically consider the limit of F ǫ as ǫ → 0, where � � � F ǫ ( u , h , Ω) = 1 |∇ u | 2 dx + ǫ |∇ u | 2 dx − fudx . (1) 2 2 Ω Σ ǫ Ω The limit is in the context of Γ -convergence. 3 / 23

Introduction Existence of Optimal Shapes Background :Optimization Problems in Thermal Insulation Stability The Asymptotic Problem Conclusion The Asymptotic Problem (Acerbi-Buttazzo 86’) F ǫ Γ -converge to � � � u 2 F ( u , h , Ω) := 1 |∇ u | 2 dx + 1 h d σ − fudx . 2 2 Ω ∂ Ω Ω To find the best distribution of insulation material of total amount m > 0 around Ω , is to find the solution of minimization problem min u ∈ H 1 (Ω) F ( u , h , Ω) min (2) h ∈H m To find the optimal shape of thermal insulation body with prescribed volume V 0 > 0, is to find the solution of inf { min u ∈ H 1 (Ω) F ( u , h , Ω) : | Ω | = V 0 } min (3) h ∈H m 4 / 23

Introduction Existence of Optimal Shapes Background :Optimization Problems in Thermal Insulation Stability The Asymptotic Problem Conclusion (2) is equivalent to inf { J ( u , Ω) : u ∈ H 1 (Ω) } , where � �� � 2 � J ( u , Ω) := 1 |∇ u | 2 dx + 1 | u | d H n − 1 − fudx 2 2 m Ω ∂ Ω Ω (4) To find the optimal shape of thermal insulation body with prescribed volume V 0 > 0, is to find the solution of � � J ( u , Ω) : u ∈ H 1 (Ω) , | Ω | = V 0 inf (5) 5 / 23

Introduction Existence of Optimal Shapes Background :Optimization Problems in Thermal Insulation Stability The Asymptotic Problem Conclusion (Bucur-Buttazzo-Nitsch 16’) Fix any domain Ω , J ( u , Ω) admits a unique minimizer u Ω ∈ H 1 (Ω) . (Bucur-Buttazzo-Nitsch 16’) If Ω = B R and f ≡ 1, then u Ω = R 2 −| x | 2 m + n 2 w n R n − 2 . 2 n (Bucur-Buttazzo-Nitsch 16’) B R is stationary shape when f ≡ 1. 6 / 23

Introduction Existence of Optimal Shapes Background :Optimization Problems in Thermal Insulation Stability The Asymptotic Problem Conclusion Questions : (Bucur-Buttazzo-Nitsch 17’) : Can the infimum of (5) be attained at a pair ( u , Ω) ? If so, is B R an optimal shape for some R > 0? 7 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion M -conformal domain Definition We say Ω is M -conformal domain if there is a bi-Lipschitz map Φ with constant M such that Φ( B ) = Ω and Φ( ∂ B ) = ∂ Ω . Remark If Ω is a M-conformal domain, then P (Ω) ≤ M n − 1 H n − 1 ( ∂ B ) = C ( M , n ) . Example Ω ⊂ B R 0 , convex and | Ω | = V 0 > 0, M = M ( n , V 0 , R 0 ) . Ω is uniformly star-shaped, M depends on the star-shaped constant. 8 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion If Ω is M -conformal and u ∈ H 1 (Ω) , then Trace inequality || u || L 2 ( ∂ Ω , d σ ) ≤ C ( M , n ) || u || H 1 (Ω) , (6) Poincaré inequality �� � 2 � �� � u 2 ≤ C ( M , n ) |∇ u | 2 dx + | u | d H n − 1 , (7) Ω Ω ∂ Ω Extension property || Eu || H 1 ( R n ) ≤ C ( n , M ) || u || H 1 (Ω) . (8) 9 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion Lower semicontinuity (7) and (8) allows us to carry out direct method, in the sense that if ( u i , Ω i ) is minimizing sequence realizing (4), then it follows that u i as extension funtions are bounded in H 1 ( R n ) and thus have H 1 weak limit u . It is clear we can find M -conformal domain Ω as L 1 limit of Ω i , then � � |∇ u | 2 ≤ lim inf |∇ u i | 2 (9) i →∞ Ω Ω i follows by showing ∇ u i χ Ω i ⇀ u χ Ω weakly in L 2 ( R n ) . (6) together with weak convergence argument gives � � | u | ≤ lim inf | u | . (10) i →∞ ∂ Ω ∂ Ω i 10 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion Existence of optimal shape in energy problem Theorem Given f ∈ L 2 ( R n ) and m , M are fixed positive constants, then the infimum of �� � 2 � � J ( u , Ω) := 1 |∇ u | 2 dx + 1 | u | d H n − 1 − fu (11) 2 2 m Ω ∂ Ω Ω can be attained over all u ∈ H 1 (Ω) and M-conformal domain Ω in B R with | Ω | = V 0 > 0 . 11 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion Flow Map Definition Let η be a smooth vector field, and F t ( x ) := F ( t , x ) solve the ODE � d dt F ( t , x ) = η ( F ( t , x )) (12) F 0 ( x ) = x . We call F t is the flow map generated by η . JF t = 1 + t div η + t 2 � ∇ (div η ) · η + (div η ) 2 � + O ( t 3 ) 2 ( ∇ x F t ( x )) − 1 = I − t ∇ x η ( x )+ t 2 � � ( ∇ x η ( x )) 2 − ∇ 2 + O ( t 3 ) x η ( x ) · η ( x ) 2 where � � ∇ 2 η · η = η i jk η k . ent ij 12 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion � Ω div η = 0. u = u Ω is the unique minimizer of Always assume inf u ∈ H 1 (Ω) J ( u , Ω) Definition We say Ω is a stationary solution to (4) if for any smooth variation vector field η , d dt J ( u Ω ◦ F − 1 , F t (Ω)) = 0 , t where F t is the flow map generated by η . 13 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion Definition We say Ω is stable under smooth variation vector field η if d 2 dt 2 J ( u Ω ◦ F − 1 , F t (Ω)) ≥ 0 , t where F t is the flow map generated by η . Definition We say Ω is a stable shape if Ω is stable under any smooth variation vector fields. 14 / 23

Introduction Existence of Optimal Shapes Direct Approach Stability First Variation and Stationary Condition Conclusion First Variation and Stationary Condition Let I ( t ) = J ( u t , Ω t ) = I 1 ( t ) + I 2 ( t ) , where u t := u ◦ F − 1 , t Ω t := F t (Ω) , � � I 1 ( t ) := 1 |∇ u t | 2 − u 2 Ω t Ω t and �� � 2 1 I 2 ( t ) = u t . 2 m ∂ Ω t � 2 � � � � � � I ′ ( 0 ) = 1 | ∂ τ u | 2 − | ∂ ν u | 2 − 2 u + u uH ( η · ν ) 2 m ∂ Ω ∂ Ω ∂ Ω (13) Therefore, we obtain the following stationary condition : � 2 � � � | ∂ τ u | 2 − | ∂ ν u | 2 − 2 u + u uH ≡ const . (14) m ∂ Ω ∂ Ω 15 / 23

Introduction Existence of Optimal Shapes Second Variation and Stability Stability Conclusion Second Variation of Surface Energy � � d 2 u t = d 2 � | g ( t ) | d ˜ u ◦ ψ x dt 2 dt 2 ∂ Ω t � = d � | g ( t ) | ] ◦ F t ◦ ψ d ˜ u ◦ ψ [ ζ H x dt � �� � � � ζ 2 H 2 + ζ ( − ∆ ∂ Ω t ζ − | A | 2 ζ ) = u ◦ ψ | g | ◦ F t ◦ ψ � + u ◦ ψ ˙ | g | ◦ F t ◦ ψ d ˜ ζ H x , where ˙ ζ := d dt ( ζ ◦ F t ◦ ψ ) can be calculated easily as ζ = d ˙ dt ( η · ν ) = �∇ ηη, ν � + � η, ˙ ν � = �∇ ηη, ν � , since η ⊥ ∂ Ω 16 / 23

Introduction Existence of Optimal Shapes Second Variation and Stability Stability Conclusion Second Variation of Surface Energy When Ω = B R , � � 2 ( 0 ) = 1 � � |∇ ∂ Ω ζ | 2 + ( H 2 − | A | 2 ) ζ 2 + H �∇ ηη, ν � I ′′ u u m ∂ B R ∂ B R � m |∇ ∂ B R ζ | 2 + ( n − 1 )( n − 2 ) ζ 2 + n − 1 = �∇ ηη, ν � n 3 ω n R n − 3 R 2 R ∂ B R Notice �∇ ηη, ν � = ζ div η − ζ 2 H , hence �� � m S n − 1 |∇ S n − 1 ζ | 2 − ( n − 1 ) ζ 2 I ′′ 2 ( 0 ) = n 3 w n R n − 1 � �� � : A 2 � + n − 1 u ( η · x ) div η nR ∂ B R 17 / 23