GoBack Ellipticity and Fredholm boundary value problems Katya - PowerPoint PPT Presentation

GoBack

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela Department of Mathematics, University of Joensuu, Finland January 4, 2006

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Elliptic PDEs Main Result The SL-condition Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 2/22

Elliptic Systems General linear q ’th order PDE: Elliptic PDEs Classical Definition � Problem a µ ( x ) ∂ µ y = f Ay = DN-elliptic PDEs Involutive PDEs | µ |≤ q Main Result where x ∈ Ω ⊂ R n , a µ ( x ) ∈ R k × m , µ ∈ N n 0 and k ≥ m . The SL-condition Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 3/22

Elliptic Systems General linear q ’th order PDE: Elliptic PDEs Classical Definition � Problem a µ ( x ) ∂ µ y = f Ay = DN-elliptic PDEs Involutive PDEs | µ |≤ q Main Result where x ∈ Ω ⊂ R n , a µ ( x ) ∈ R k × m , µ ∈ N n 0 and k ≥ m . The SL-condition Principal symbol of A is � a µ ( x ) ξ µ A = | µ | = q ξ ∈ T ∗ Ω , ξ µ = ξ µ 1 1 . . . ξ µ n n Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 3/22

Elliptic Systems General linear q ’th order PDE: Elliptic PDEs Classical Definition � Problem a µ ( x ) ∂ µ y = f Ay = DN-elliptic PDEs Involutive PDEs | µ |≤ q Main Result where x ∈ Ω ⊂ R n , a µ ( x ) ∈ R k × m , µ ∈ N n 0 and k ≥ m . The SL-condition Principal symbol of A is � a µ ( x ) ξ µ A = | µ | = q ξ ∈ T ∗ Ω , ξ µ = ξ µ 1 1 . . . ξ µ n n Definition. The differential operator A is called elliptic in Ω , if A is injective for all real ξ � = 0 and all x ∈ Ω . Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 3/22

Example Example. Consider the transformation of the two-dimensional Elliptic PDEs Classical Definition Laplace equation ∆ u = u 20 + u 02 = 0 to the first order system Problem DN-elliptic PDEs 10 − y 2 = 0 ,    y 1 ξ 1 0 0 Involutive PDEs   01 − y 3 = 0 , Main Result y 1 Ly = L =  . ξ 2 0 0     y 2 10 + y 3 01 = 0 . 0 ξ 1 ξ 2  The SL-condition The transformed system is not elliptic, although it is equivalent to Laplace’s equation. Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 4/22

Example Example. Consider the transformation of the two-dimensional Elliptic PDEs Classical Definition Laplace equation ∆ u = u 20 + u 02 = 0 to the first order system Problem DN-elliptic PDEs 10 − y 2 = 0 ,    y 1 ξ 1 0 0 Involutive PDEs   01 − y 3 = 0 , Main Result y 1 Ly = L =  . ξ 2 0 0     y 2 10 + y 3 01 = 0 . 0 ξ 1 ξ 2  The SL-condition The transformed system is not elliptic, although it is equivalent to Laplace’s equation. 1st approach to resolve this issue consists in the introduction of a weighted symbol (Douglis and Nirenberg, 1955) Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 4/22

Example Example. Consider the transformation of the two-dimensional Elliptic PDEs Classical Definition Laplace equation ∆ u = u 20 + u 02 = 0 to the first order system Problem DN-elliptic PDEs 10 − y 2 = 0 ,    y 1 ξ 1 0 0 Involutive PDEs   01 − y 3 = 0 , Main Result y 1 Ly = L =  . ξ 2 0 0     y 2 10 + y 3 01 = 0 . 0 ξ 1 ξ 2  The SL-condition The transformed system is not elliptic, although it is equivalent to Laplace’s equation. 1st approach to resolve this issue consists in the introduction of a weighted symbol (Douglis and Nirenberg, 1955) s i : weight for the i th equation t j : weight for the j th dependent variable s i + t j ≥ q ij Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 4/22

DN-elliptic systems of PDEs The weighted (principal) symbol of A is Elliptic PDEs Classical Definition i,j ξ µ . Problem � � � � � i,j = a µ ( x ) A w DN-elliptic PDEs Involutive PDEs | µ | = s i + t j Main Result The SL-condition Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs The weighted (principal) symbol of A is Elliptic PDEs Classical Definition i,j ξ µ . Problem � � � � � i,j = a µ ( x ) A w DN-elliptic PDEs Involutive PDEs | µ | = s i + t j Main Result Definition. A is DN–elliptic, if A w is injective for all real ξ � = 0 and The SL-condition for all x ∈ Ω . Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs The weighted (principal) symbol of A is Elliptic PDEs Classical Definition i,j ξ µ . Problem � � � � � i,j = a µ ( x ) A w DN-elliptic PDEs Involutive PDEs | µ | = s i + t j Main Result Definition. A is DN–elliptic, if A w is injective for all real ξ � = 0 and The SL-condition for all x ∈ Ω . Ordinary ellipticity is a special case of DN-ellipticity with weights s i = 0 and t j = q . Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs The weighted (principal) symbol of A is Elliptic PDEs Classical Definition i,j ξ µ . Problem � � � � � i,j = a µ ( x ) A w DN-elliptic PDEs Involutive PDEs | µ | = s i + t j Main Result Definition. A is DN–elliptic, if A w is injective for all real ξ � = 0 and The SL-condition for all x ∈ Ω . Ordinary ellipticity is a special case of DN-ellipticity with weights s i = 0 and t j = q . Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs The weighted (principal) symbol of A is Elliptic PDEs Classical Definition i,j ξ µ . Problem � � � � � i,j = a µ ( x ) A w DN-elliptic PDEs Involutive PDEs | µ | = s i + t j Main Result Definition. A is DN–elliptic, if A w is injective for all real ξ � = 0 and The SL-condition for all x ∈ Ω . Ordinary ellipticity is a special case of DN-ellipticity with weights s i = 0 and t j = q . L is DN-elliptic: s 1 = s 2 = − 1 , s 3 = 0 , t 1 = 2 , t 2 = t 3 = 1   ξ 1 − 1 0 L w = ξ 2 0 − 1  .    0 ξ 1 ξ 2 Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

Involutive Systems of PDEs 2nd approach: to complete a system to the involutive form and check ordinary ellipticity Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs 2nd approach: to complete a system to the involutive form and check ordinary ellipticity Elliptic PDEs Classical Definition Problem If Ly = 0 , then y 2 01 − y 3 10 = 0 . This new equation is called a DN-elliptic PDEs differential consequence or integrability condition of the initial Involutive PDEs Main Result system. The SL-condition Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs 2nd approach: to complete a system to the involutive form and check ordinary ellipticity Elliptic PDEs Classical Definition Problem If Ly = 0 , then y 2 01 − y 3 10 = 0 . This new equation is called a DN-elliptic PDEs differential consequence or integrability condition of the initial Involutive PDEs Main Result system. The SL-condition 10 − y 2 = 0 ,    y 1 ξ 1 0 0   01 − y 3 = 0 ,  y 1 ξ 2 0 0    L ′ = L ′ y = ,  .   y 2 10 + y 3 01 = 0 ,  0 ξ 1 ξ 2      y 2 01 − y 3 10 = 0 . 0 ξ 2 − ξ 1  Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs 2nd approach: to complete a system to the involutive form and check ordinary ellipticity Elliptic PDEs Classical Definition Problem If Ly = 0 , then y 2 01 − y 3 10 = 0 . This new equation is called a DN-elliptic PDEs differential consequence or integrability condition of the initial Involutive PDEs Main Result system. The SL-condition 10 − y 2 = 0 ,    y 1 ξ 1 0 0   01 − y 3 = 0 ,  y 1 ξ 2 0 0    L ′ = L ′ y = ,  .   y 2 10 + y 3 01 = 0 ,  0 ξ 1 ξ 2      y 2 01 − y 3 10 = 0 . 0 ξ 2 − ξ 1  L ′ is the involutive form of L because no more new first order differential consequences can be found. Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22