Weak Solutions for a Degenerate Elliptic Dirichlet Problem Aurelian - PowerPoint PPT Presentation

Weak Solutions for a Degenerate Elliptic Dirichlet Problem Aurelian Gheondea Bilkent University, Ankara IMAR, Bucharest Spectral Problems for Operators and Matrices The Third Najman Conference Biograd, 18th of September, 2013 Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 1 / 25

Branko Najman (1946–1996) Picture taken by G.M. Bergmann at Oberwolfach in 1980. http://owpdb.mfo.de/detail?photo id=5675 Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 2 / 25

Outline Triplets of Hilbert Spaces 1 Closely Embedded Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces A Dirichlet Problem Associated to a Class of Degenerate Elliptic PDE 2 Spaces The Assumptions The Main Result Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 3 / 25

Triplets of Hilbert Spaces Closely Embedded Hilbert Spaces Closely Embedded Hilbert Spaces Let H and H + be two Hilbert spaces. The Hilbert space H + is called closely embedded in H if: (ce1) There exists a linear manifold D ⊆ H + ∩ H that is dense in H + . (ce2) The embedding operator j + with domain D is closed, as an operator H + → H . Axiom (ce1) means that on D the algebraic structures of H + and H agree. Axiom (ce2) means that the operator j + with Dom( j + ) = D ⊆ H + defined by j + x = x ∈ H , for all x ∈ D , is closed. Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 4 / 25

Triplets of Hilbert Spaces Closely Embedded Hilbert Spaces The Kernel Operator Let H + be a Hilbert space that is closely embedded in H , and let j + + ∈ C ( H ) + and denote the corresponding closed embedding. Then A = j + j ∗ � j + h , k � = � h , Ak � + , h ∈ Dom( j + ) , k ∈ Dom( A ) , (2.1) more precisely, A has the range in H + and it can also be viewed as the adjoint of the embedding j + . The operator A is called the kernel operator associated to the closed embedding of H + in H . L. Schwartz — for continuous embeddings Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 5 / 25

Triplets of Hilbert Spaces Closely Embedded Hilbert Spaces The Space R ( T ) Let T ∈ C ( G , H ) be a closed and densely defined linear operator, where G is another Hilbert space. On Ran( T ) we consider a new inner product � Tu , Tv � T = � u , v � G , (2.2) where u , v ∈ Dom( T ) ⊖ Ker( T ). With respect to this new inner product Ran( T ) can be completed to a Hilbert space that we denote by R ( T ), closely embedded in H , and in such a way that j T : R ( T ) → H has the property that j T j ∗ T = TT ∗ . Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 6 / 25

Triplets of Hilbert Spaces Closely Embedded Hilbert Spaces The Space D ( T ) Let T ∈ C ( H , G ) with Ker( T ) a closed subspace of H . Define the norm | x | T := � Tx � G , x ∈ Dom( T ) ⊖ Ker( T ) , (2.3) and let D ( T ) be the Hilbert space completion of the pre-Hilbert space Dom( T ) ⊖ Ker( T ) with respect to the norm | · | T associated the inner product ( · , · ) T ( x , y ) T = � Tx , Ty � G , x , y ∈ Dom( T ) ⊖ Ker( T ) . (2.4) Define i T from D ( T ) and valued in H by i T x := x , x ∈ Dom( i T ) = Dom( T ) ⊖ Ker( T ) . (2.5) The operator i T is closed and D ( T ) is closely embedded in H , with the underlying closed embedding i T . The operator Ti T admits a unique isometric extension � T : D ( T ) → G . Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 7 / 25

Triplets of Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces By definition, ( H + ; H ; H − ) is called a triplet of closely embedded Hilbert spaces if: (th1) H + is a Hilbert space closely embedded in the Hilbert space H , with the closed embedding denoted by j + , and such that Ran( j + ) is dense in H . (th2) H is closely embedded in the Hilbert space H − , with the closed embedding denoted by j − , and such that Ran( j − ) is dense in H − . (th3) Dom( j ∗ + ) ⊆ Dom( j − ) and for every vector y ∈ Dom( j − ) ⊆ H we have � |� x , y � H | � � y � − = sup | x ∈ Dom( j + ) , x � = 0 . � x � + The kernel operator A = j + j ∗ + is a positive selfadjoint operator in H that is one-to-one. Then, H = A − 1 is a positive selfadjoint operator in H and it is called the Hamiltonian of the triplet. Note that, as a consequence of (th3), we actually have Dom( j ∗ + ) = Dom( j − ). Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 8 / 25

Triplets of Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces Generation of Triplets of Hilbert Spaces: Factoring the Hamiltonian Theorem Let H be a positive selfadjoint operator in the Hilbert space H , that admits an inverse A = H − 1 , possibly unbounded. Then there exists T ∈ C ( H , G ) , with Ran( T ) dense in G and H = T ∗ T. In addition, let S = T − 1 ∈ C ( G , H ) . Then: (i) The Hilbert space H + := D ( T ) := R ( S ) is closely embedded in H with its embedding i T having range dense in H , and its kernel operator A = i T i ∗ T coincides with H − 1 . (ii) H is closely embedded in the Hilbert space H − = R ( T ∗ ) with its embedding j − 1 T ∗ having range dense in R ( T ∗ ) . The kernel operator B = j − 1 T ∗ j − 1 ∗ of this embedding is unitary equivalent with A = H − 1 . T ∗ Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 9 / 25

Triplets of Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces Generation of Triplets of Hilbert Spaces: Weak Solutions Theorem (continued) (iii) The operator V = i ∗ T | Ran( T ∗ ) , that is, x ∈ Dom( T ) , y ∈ Ran( T ∗ ) , � i T x , y � H = ( x , Vy ) T , (2.6) extends uniquely to a unitary operator � V between the Hilbert spaces R ( T ∗ ) and D ( T ) . (iv) The operator H, when viewed as a linear operator with domain dense in D ( T ) and range in R ( T ∗ ) , extends uniquely to a unitary operator H : D ( T ) → R ( T ∗ ) , and � � H = � V − 1 . Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 10 / 25

Triplets of Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces Generation of Triplets of Hilbert Spaces: Dual Space Theorem (continued) (v) The operator Θ: R ( T ∗ ) → D ( T ) ∗ defined by (Θ α )( x ) := ( � α ∈ R ( T ∗ ) , x ∈ D ( T ) , V α, x ) T , (2.7) provides a canonical and unitary identification of the Hilbert space R ( T ∗ ) with the conjugate space D ( T ) ∗ , in particular, for all y ∈ Dom( T ∗ ) � |� y , x � H | � � y � T ∗ = sup | x ∈ Dom( T ) \ { 0 } (2.8) . | x | T Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 11 / 25

� � � � � � � � Triplets of Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces Generation of Triplets of Hilbert Spaces: The General Picture H � � � ����������� V j − 1 � R ( T ∗ ) = H − i T � H T ∗ H + = D ( T ) j T ∗ � ����������� i ∗ T H = A − 1 V = e e H = e e A A − 1 A i − 1 � H T D ( T ) D ( T ) = H + i T i T H Hamiltonian Berezansky — continuous embeddings A = H − 1 Kernel Operator H = T ∗ T Factor Operator A = SS ∗ Factor Operator Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 12 / 25

A Dirichlet Problem for a Degenerate Elliptic PDE Spaces The Gradient Let Ω be an open (nonempty) set of the R N . Let D j = i ∂ ∂ x j , ( j = 1 , . . . , N ) be the operators of differentiation with respect to the coordinates of points x = ( x 1 , . . . , x N ) in R N . For a multi-index + , let x α = x α 1 N , D α = D α 1 1 · · · x α N 1 · · · D α N α = ( α 1 , . . . , α N ) ∈ Z N N . ∇ l = ( D α ) | α | = l denotes the gradient of order l , where l is a fixed nonnegative integer. Letting m = m ( N , l ) denote the number of all multi-indices α = ( α 1 , . . . , α N ) such that | α | = α 1 + · · · + α N = l , ∇ l can be viewed as an operator acting from L 2 (Ω) into L 2 (Ω; C m ) defined on its maximal domain, the Sobolev space W l 2 (Ω), by ∇ l u = ( D α u ) | α | = l , u ∈ W l 2 (Ω) . Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 13 / 25

A Dirichlet Problem for a Degenerate Elliptic PDE Spaces The Underlying Spaces W l 2 (Ω) consists of those functions u ∈ L 2 (Ω) whose distributional derivatives D α u belong to L 2 (Ω) for all α ∈ Z N + , | α | ≤ l and with norm � � � 1 / 2 � D α u � 2 � u � W l 2 (Ω) = , (3.1) L 2 (Ω) | α |≤ m W l 2 (Ω) becomes a Hilbert space that is continuously embedded in L 2 (Ω). l ◦ 2 (Ω) denotes the closure of C ∞ 0 (Ω) in the space W l 2 (Ω). W Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 14 / 25