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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
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