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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

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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

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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

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Outline

1

Triplets of Hilbert Spaces Closely Embedded Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces

2

A Dirichlet Problem Associated to a Class of Degenerate Elliptic PDE Spaces The Assumptions The Main Result

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 3 / 25

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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

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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+ denote the corresponding closed embedding. Then A = j+j∗

+ ∈ C(H)+ and

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

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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, TvT = u, vG, (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 jT : R(T) → H has the property that jTj∗

T = TT ∗.

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 6 / 25

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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 := TxG, 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, TyG, x, y ∈ Dom(T) ⊖ Ker(T). (2.4) Define iT from D(T) and valued in H by iTx := x, x ∈ Dom(iT) = Dom(T) ⊖ Ker(T). (2.5) The operator iT is closed and D(T) is closely embedded in H, with the underlying closed embedding iT. The operator TiT admits a unique isometric extension T : D(T) → G.

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 7 / 25

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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

y− = sup |x, yH| x+ | x ∈ Dom(j+), x = 0

  • .

The kernel operator A = j+j∗

+ is a positive selfadjoint operator in H that is

  • ne-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

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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 iT having range dense in H, and its kernel

  • perator A = iTi∗

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∗ T ∗

  • f this embedding is unitary equivalent with A = H−1.

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 9 / 25

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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,

iTx, yH = (x, Vy)T, x ∈ Dom(T), y ∈ Ran(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

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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 (Θα)(x) := ( V α, x)T, α ∈ R(T ∗), x ∈ D(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 ∗) yT ∗ = sup |y, xH| |x|T | x ∈ Dom(T) \ {0}

  • .

(2.8)

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 11 / 25

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Triplets of Hilbert Spaces Triplets of Closely Embedded Hilbert Spaces

Generation of Triplets of Hilbert Spaces: The General Picture

H

V

  • H+ = D(T)

iT

H

H=A−1

  • i∗

T

  • jT∗

R(T ∗) = H−

j−1

T∗

  • e

V =e A

  • D(T)

iT

H

iT

  • A
  • D(T) = H+

e H=e A−1

  • i−1

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

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A Dirichlet Problem for a Degenerate Elliptic PDE Spaces

The Gradient

Let Ω be an open (nonempty) set of the RN. Let Dj = i ∂

∂xj ,

(j = 1, . . . , N) be the operators of differentiation with respect to the coordinates of points x = (x1, . . . , xN) in RN. For a multi-index α = (α1, . . . , αN) ∈ ZN

+, let xα = xα1 1 · · · xαN N , Dα = Dα1 1 · · · DαN N .

∇l = (Dα)|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 = (α1, . . . , αN) such that |α| = α1 + · · · + αN = l, ∇l can be viewed as an operator acting from L2(Ω) into L2(Ω; Cm) defined on its maximal domain, the Sobolev space W l

2(Ω), by

∇lu = (Dαu)|α|=l, u ∈ W l

2(Ω).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 13 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE Spaces

The Underlying Spaces

W l

2(Ω) consists of those functions u ∈ L2(Ω) whose distributional

derivatives Dαu belong to L2(Ω) for all α ∈ ZN

+, |α| ≤ l and with norm

uW l

2(Ω) =

|α|≤m

Dαu2

L2(Ω)

1/2 , (3.1) W l

2(Ω) becomes a Hilbert space that is continuously embedded in L2(Ω).

  • W

l 2 (Ω) denotes the closure of C ∞ 0 (Ω) in the space W l 2(Ω).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 14 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE Spaces

More Spaces

The space

  • L

l p (Ω), (1 ≤ p < ∞) is defined as the completion of C ∞ 0 (Ω)

under the metric corresponding to up,l := ∇luLp(Ω) =

  • Ω

|α|=l

|Dαu(x)|2 p/2 d x 1/p , u ∈ C ∞

0 (Ω).

The elements of

  • L

l p (Ω) can be realized as locally integrable functions on

Ω vanishing at the boundary ∂Ω and having distributional derivatives of

  • rder l in Lp(Ω).

Moreover, these functions, after modification on a set of zero measure, are absolutely continuous on every line which is parallel to the coordinate axes.

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 15 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Assumptions

The Principal Symbol

On Ω there is defined an m × m matrix valued measurable function a, more precisely, a(x) = [aαβ(x)], |α|, |β| = l, x ∈ Ω, where the scalar valued functions aα,β are measurable on Ω for all multi-indices |α|, |β| = l. (C1) For almost all (with respect to the n-dimensional standard Lebesgue measure) x ∈ Ω, the matrix a(x) is nonnegative (positive semidefinite), that is,

  • |α|,|β|=l

aαβ(x)ηβηα ≥ 0, for all η = (ηα)|α|=l ∈ Cm. According to the condition (C1), there exists an m × m matrix valued measurable function b on Ω, such that a(x) = b(x)∗b(x), for almost all x ∈ Ω, where b(x)∗ denotes the Hermitian conjugate matrix of the matrix b(x).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 16 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Assumptions

Conditions

(C2) There is a nonnegative measurable function c on Ω such that, for almost all x ∈ Ω and all ξ = (ξ1, . . . , ξN) ∈ CN, |b(x) ξ| ≥ c(x)|˜ ξ|, where ˜ ξ = (ξα)|α|=l is the vector in Cm with ξα = ξα1

1 ... ξαN N .

(C3) All the entries bαβ of the m × m matrix valued function b are functions in L1,loc(Ω). (C4) The function c in (C2) has the property that 1

  • c ∈ L2(Ω).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 17 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Assumptions

The Operator T

Under the conditions (C1)–(C4), we consider the operator T acting from L2(Ω) to L2(Ω; Cm) and defined by (Tu)(x) = b(x)∇lu(x), for almost all x ∈ Ω, (3.2)

  • n its domain

Dom(T) = {u ∈

  • W

l 2 (Ω) | b∇lu ∈ L2(Ω; Cm)}.

(3.3)

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 18 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Assumptions

The Problem

Our aim is to describe, in view of the abstract model, the triplet of closely embedded Hilbert spaces (D(T); L2(Ω); R(T ∗)) associated with the

  • perator T defined at (3.2) and (3.3).

In terms of these results, we obtain information about weak solutions for the corresponding operator equation involving the Hamiltonian operator H = T ∗T of the triplet, which in fact is a Dirichlet boundary value problem in L2(Ω) with homogeneous boundary values.

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 19 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Assumptions

The Problem

This problem is associated to the differential sesqui-linear form a[u, v] =

  • Ω

a(x)∇l(x), ∇l(x) d x (3.4) =

  • |α|=|β|=l
  • Ω

aαβ(x)Dβu(x)Dαv(x)dx, u, v ∈ C ∞

0 (Ω),

which, as will be seen, can be extended up to elements of D(T). The problem can be reformulated as follows : given f ∈ D(T)∗ (which is canonically identified withe R(T ∗)), find v ∈ D(T) such that a[u, v] = u, f for all u ∈ D(T), (3.5) where ·, · denotes the duality between D(T) and D(T)∗. The problem in (3.5) can be considered only for u ∈

  • W

l 2 (Ω), or, even

more restrictively, only for u ∈ C ∞

0 (Ω).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 20 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Main Result

The Main Result

Theorem For Ω a domain in RN and l ∈ N, let a(x) = [aαβ(x)] = b(x)∗b(x), |α|, |β| = l, x ∈ Ω, satisfy the conditions (C1)–(C4), and consider the differential sesqui-linear form a[u, v] =

  • Ω

a(x)∇l(x), ∇l(x) d x =

  • |α|=|β|=l
  • Ω

aαβ(x)Dβu(x)Dαv(x) d x, u, v ∈ C ∞

0 (Ω),

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 21 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Main Result

The Main Result

Theorem (Continuation) Then: (1) The operator T acting from L2(Ω) to L2(Ω; Cm) and defined by (Tu)(x) = b(x)∇lu(x) for x ∈ Ω and u ∈ Dom(T) = {u ∈

  • W

l 2 (Ω) | b∇lu ∈ L2(Ω; Cm)} is closed, densely

defined, and injective. (2) The pre-Hilbert space Dom(T) with norm |u|T = (

  • Ω |b(x)∇lu(x)|2 d x)

1 2 , has a unique Hilbert space completion,

denoted by Hl

a(Ω), that is continuously embedded into

  • L

l 1 (Ω).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 22 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Main Result

The Main Result

Theorem (Continuation) (3) The conjugate space of

  • H

l a(Ω), denoted by

  • H

−l a (Ω), can be

realized in such a way that, for any f ∈

  • H

−l a

(Ω) there exist elements g ∈ L2(Ω; Cm) such that f (u) =

  • Ω

g(x), b(x)∇lu(x) d x, u ∈

  • W

l 2 (Ω),

(3.6) and f ◦

H

−l a (Ω) = inf{gL2(Ω;Cm) | g ∈ L2(Ω; Cm) such that (3.6) holds }. Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 23 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Main Result

The Main Result

Theorem (Continuation) (4) (

  • H

l a(Ω); L2(Ω);

  • H

−l a (Ω)) is a triplet of closely embedded Hilbert

spaces. (5) For every f ∈

  • H

−l a (Ω) there exists a unique v ∈ Hl a(Ω) that solves

the Dirichlet problem associated to the sesquilinear form a, in the sense that a[u, v] = u, f for all u ∈ Hl

a(Ω).

More precisely, v = H−1f , where H is the unitary operator acting between

  • H

l a (Ω) and

  • H

−l a

(Ω) that uniquely extends the positive selfadjoint

  • perator H = T ∗T in L2(Ω).

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 24 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Main Result

The Paper

  • P. Cojuhari, A. Gheondea: Triplets of Hilbert spaces,

arXiv:1309.0176.

  • P. Cojuhari, A. Gheondea, Triplets of Dirichlet type spaces on the

unit polydisc, Complex Analysis and Operator Theory, (to appear). DOI 10.1007/s11785-012-0269-z

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 25 / 25

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A Dirichlet Problem for a Degenerate Elliptic PDE The Main Result

Yu.M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence RI 1968. Yu.M. Berezansky, Y.G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Mathematical Physics and Applied Mathematics, Kluwer Academic Publishers, Dordrecht, 1995 P.A. Cojuhari, A. Gheondea: Closed embeddings of Hilbert spaces, J. Math. Anal. Appl. 369(2010), 60–75. I.M. Gelfand, A.G. Kostychenko, On eigenfunction expansions

  • f differenital an other operators [Russian], Dokl. Akad. Nauk SSSR,

103(1955), 349-352. I.M. Gelfand, N.Y. Vilenkin, Generalized Functions, Vol. IV, Academic Press, New York 1964.

  • L. Schwartz: Sous espace Hilbertiens d’espaces vectoriel

topologiques et noyaux associ´ es (noyaux reproduisants), J. Analyse Math., 13(1964), 115–256.

Aurelian Gheondea (BU & IMAR) Weak Solutions Biograd, September 18, 2013 25 / 25