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Form Methods for Evolution Equations, and Applications

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The purpose of this lecture is to introduce the concept of C0 semigroups and their generators and to derive some basic properties.

Motivation

Definition and some basic properties

Let D⊆X be a dense subset of X and assume that for all x∈D the sequence (Bnx)n∈N is convergent. It is easy to see that S(t) leaves this subspace invariant for all t>0 and that this implies that the restriction of T from S to this subspace is a C0-semigroup.

Interlude: operators, integration and differentiation

Operators

Again, we can see that T is a one-parameter semigroup (not a group), and the angle in (a) shows that T is a C0-semigroup. Let's mention that in most cases of using the sum of two operators, the domain of one of the operators is a subset of the domain of the other, or, more precisely, one of the operators is defined everywhere.

Integration of continuous functions

This means that there exists a unique continuous extension of the integral to functions in the closure T([a, b];X) in the Banach space. Moreover, the expansion of the integral is linear and the inequality (1.1) carries over to all f ∈R([a, b];X).

Resolvent set, spectrum and resolvent

Then the inverse of (λI−A)−1 is a closed operator that is defined on all X. d) Usually, in the discussion of operator theory, the above terms are defined only for the case of complex Banach spaces. This means that the solvent norm must explode as λ approaches σ(A). b) As in the analysis of the first year, the analyticity of R(·, A) implies that R(·, A) is infinitely differentiable, and we can take derivatives from the power series (2.1).

Integration of operator valued functions, and improper integrals . 15

This means that it is sufficient to prove the existence and formula of the solvent for the case λ= 0 and ω <0. Next, we must show that the necessary conditions for the generator are also sufficient.

An exponential formula

Now the proof is more or less complete as in part (ii) of the proof of Theorem 2.9. And in section 3.4 the special case of holomorphic semigroups in Hilbert spaces will be treated.

Holomorphic semigroups

The local uniform equicontinuity of the sequence (fn) together with the pointwise convergence implies that (fn) converges uniformly to f. In view of the continuity at 0 of the restriction of T(·)Ax to Σθ0,0 one thus obtains the assertion. In view of Theorem 3.13, it is justified to call the generator of the C0-semigroup T [0,∞ ) also the generator of the holomorphic C0-semigroupT.

Generation of contractive holomorphic semigroups

It also follows from the hypotheses and Theorem 2.9 that for every α∈(−θ, θ) the operator eiαA generates a contractiveC0 semigroup; call it Ta. From Corollary 3.8 we deduce that T is holomorphic on Σθ and from Theorem 2.12 that T [0,∞) is the C0 semigroup generated by A.

The Lumer-Phillips theorem

  • Convolution
  • Distributional derivatives
  • Definition of H 1 (Ω)
  • Denseness properties

Moreover, Cc∞(Ω) :=C∞(Ω)∩Cc(Ω) is the space of infinitely differentiable functions with compact support. i) The integral exists because ρ is bounded and has compact support. In particular, if ∂αf ∈ L1,loc(Ω), then we call ∂αf the distributional (or generalized or weak) derivative of f.

The Hilbert space method for the solution of inhomogeneous problems, and

If we move the first term on the right side to the left and use Lemma 4.17, we conclude that −∆u=f−u in the distributional sense. The uniqueness of u is a consequence of the uniqueness in the Riesz-Fr´echet representation theorem.

The Dirichlet Laplacian

In section 4.1 we collected some basic principles of Sobolev spaces, if we need them, and we use them in the following. The shape is therefore linear in the first variable, while for the second variable only half of the linearity conditions are met.

Representation theorems

It is not difficult to show that the boundary of a form equals continuity; see Exercise 5.1. If the form is symmetric, then the Lax-Milgram lemma is the same as the Riesz-Fr´echet theorem.

Semigroups by forms, the complete case

To show that A is m-accretive, we need to show the range condition ran(I+A) = H. This is a shorthand for saying that V is continuously embedded in H (abbreviated V, → H) and that V is compact in H.

The classical Dirichlet form and other examples

In that case the definition of the operator A associated with a (not mentioning the given insertion of V into H) reads as follows. One of the purposes of this lecture is to explain all the notions that occur in the previous sentence.

Adjoints of operators, and self-adjoint operators

An operator A in H is essentially called self-adjoint if it is symmetric and A is self-adjoint. Let A be a self-adjoint operator in an infinite-dimensional separable Hilbert space, and assume that there exists an orthonormal basis (en)n∈N consisting of eigenelements of A, with corresponding eigenvalues ​​(λn)n∈N.

Adjoints of forms and operators

The spectral theorem for compact self-adjoint operators

Let A be a positive self-adjoint operator with compact resolvent in an infinite-dimensional Hilbert space H. From Theorem 6.1 we know that (I + A)-1 exists in L(H), and it is easy to see that this operator is symmetric, i.e. self-adjoint.

The first version of the spectral theorem of compact self-adjoint operators is contained in [Hil06]. Gauss's theorem can be thought of as the n-dimensional version of the fundamental theorem of calculus.

The procedure used in the proof of Theorem 7.7 can be modified to obtain simultaneous approximation with respect to other properties. As for part (i) of the proof of Theorem 7.7, the continuation case is done by calling a local version of Theorem 4.3(a).

Weak normal derivative

The Neumann Laplacian

Applying Theorem 7.11, we conclude that A has compact solutions if Ω satisfies our weaker regularity property. Neumann's Laplacian Spectral Decomposition) If Ω⊆Rn is open, bounded and has continuous boundary, then ∆N has compact solver. The statement about eigenfunctions and eigenvalues ​​now follows from Theorem 6.17, except for the property that λ1 = 0.

The Robin Laplacian

Show that the numerical range num(A) lies in a parabola with vertex on the real axis and opened in the direction of the positive real axis. The Dirichlet-to-Neumann operator plays an important role in the theory of inverse problems.

Interlude: the Fredholm alternative in Hilbert space

The following interlude is a preparation for this hearing. finite-dimensional) linear algebra, it is known that the operator I2+P2KH2 is injective if and only if it is surjective. Again, using the matrix representation, we can easily see that I2+P2KH2 is surjective if and only if I+K is surjective.

Quasi-m-accretive and self-adjoint operators via compactly elliptic forms 90

This shows that S is forcing, thus invertible in L(Vj(a)), by the operator version of the Lax-Milgram lemma, Remark 5.3. Show that ∆D+m is self-adjoint, upper bounded, and has compact solvent. a) Calculate the Dirichlet-to-Neumann operator D0 for Ω = (a, b), and calculate the C0 semigroup generated by −D0.

Application to Laplacians

Let T be a contracting semigroup C0 in a Hilbert space H, let A be the generator of T, and let C 6=∅ be a closed convex subgroup of H. We note that identically the same arguments show the same properties for the Neumann Laplacian.

These properties will be shown in the next section (and the present example should serve as motivation for this treatment); see Theorem 9.14. Just like the construction of the sequence (Fk) at the beginning of the section, one can construct a sequence (Fk) that converges to F(t) :=t+, with the properties mentioned in Theorem 9.12 and such thatFk0 →1 (0,∞) pointwise.

Invariance described by forms

The three lines theorem

The content of this section is a version of the maximum principle for holomorphic functions on an unbounded set. Here and in the following we denote by k·kM the highest norm taken over by setM.

The Stein interpolation theorem

111 measure spaces the requirement on [u6= 0] in the definition of L1(Ac) can be dispensed with. The index 'c' should remind of 'compact': if Ω⊆Rn is an open set, then one can choose Ac as the system of relatively compact measurable subsets of Ω.). Finally, let L(S(Ac), L1(Ac)) denote the linear operators from S(Ac) to L1(Ac) (without continuity requirement), and let Φ : S →L(S(Ac), L1(Ac ) )) be a mapping that satisfies the following two conditions.

Interpolation of semigroups

If s∈ R, then the result proved so far can be applied to the function z 7→Φ(z+ is), and this yields the claimed inequality for general s. To show that Σθτ 3z 7→Tτ(z) is holomorphic, we use the results of Section 3.1. iii) To show the strong continuity of Tτ at 0, we use H¨older's inequality kukp.

Adjoint semigroups

Indeed, it is not difficult to show that T∗, defined by (10.2), is holomorphic. a) Using the general Hille-Yosida generation theorem (see exercise 2.4) one also obtains Theorem 10.9 for general C0-semigroups on H. More generally, if T is a one-parameter semigroup onX that is weakly continuous, then T a C0 semigroup.

Applications of invariance criteria and interpolation

Theorem 10.9 also implies that the generator A of T is self-adjoint, and since T is contractive, −A is accretive. Interestingly enough, as nice and elegant as the proof of Theorem 10.15 may seem, the angle of holomorphism for the Lp semigroup is not optimal in this case, neither for holomorphism nor for contractivity.

Elliptic operators

We define TN := TH1 and call AN := AH1 the realization of the elliptic operator A with Neumann boundary conditions. To study further properties of the semigroup onL2(Ω;R) generated by an elliptic operator with real coefficients, we need additional properties of the Sobolev space H1(Ω).

Now we show that it is even an ideal (in the sense of vector networks). Following the lines of this proof, it can be seen that, starting with a function ∈H1(Ω)+, in all steps you stay in the field of positive functions.

Elliptic operators with real coefficients

It should be noted that Theorems 10.8 and 10.15 then apply to the situations covered in Theorems 11.14 and 11.16 and yield Lp properties and holomorphic extensions for the generated semigroups.

Domination

The application of invariance criteria –– in the form of the Beurling-Deny criteria –– to symmetric elliptic operators, especially in connection with the heat equation with potential ("Schr¨odinger semigroups"), has a longer history; see for example [RS78], [Dav89]. Using Theorem 5.2 we derive from (12.1) the inequality. 12.2) This is a kind of intrinsic continuity of a that we will use throughout.

The Friedrichs extension

Then a := ˜b with dom(a) := V (using the notation of Remark 12.3(d)) is a closed sectoral form in H with the required properties, and the associated operator A is an extension of B. Then there exists a unique densely defined embedded closed sectoral form a in H such that A is associated with a.

Sectorial versus elliptic

Let A be an m-sectoral operator in H. Then there exists a unique, densely defined embedded closed sector form a in H, such that A is associated with a. Let (a, j) be a densely defined closed sector form in H. The shape ˜a is continuous for this standard and ˜-elliptic. b) 'Continuous-elliptic' implies 'closed quasi-sectoral, k·kV ∼ k·ka.

The non-complete case

Then the description of the operator associated with a is as in Theorem 12.8, but without the 'j' in conditions (a) and (c). a) Let (˜a,) be the 'termination' of (a, j) as described at the beginning of the section.˜ We point out that ˜ is generally not injective. From Corollary 12.6, we know that there exists a unique closed form a in H that is associated with A.

The Robin Laplacian for rough domains

We conclude the section with a brief introduction to the d-dimensional Hausdorff measure on Rn, for d > 0. Carath´eodory's construction of measurable sets yields a measure σd, the d-dimensional Hausdorff measure, and it turns out that Borel -sets are measurable.

The Dirichlet-to-Neumann operator for rough domains

Unfortunately, it is completely beyond the scope of the internet seminar to provide evidence for Maz'ja's inequality. The Stokes operator arises in the context of the (nonlinear!) Navier-Stokes equation and operates in a subspace of a Kn-valued L2 space.

The Stokes operator

In view of Section 13.1, the equality appearing in the description of A can be rewritten as 0. In view of (a), the condition (13.2) implies that you satisfy the 'compatibility conditions' for a vector field to be locally the gradient of a potential.

Interlude: the Bogovski˘ı formula

Note that E is a compact group (in general a proper subgroup of the convex hull of sptf ∪sptρ). ii) With the transformation variablez = x−y and thenr= 1 + |z|s in the inner integral we get Bf in the form In this form it can be differentiated under the integral to obtain Bf ∈ C∞(Rn;Kn). 13.6) Note that in the above calculation ∇ϕ satisfies the compatibility conditions, i.e.

The hypothesis (H) and the Bogovski˘ı formula

For the Calder'on-Zygmund theorem needed for this purpose, we refer to the original paper [CZ56; Theorem 2] and in [Gal11; Theorem II.11.4]. The fundamental observation for the proof is that the continuous linear operators div : H01(Ω;Kn)→L2(Ω) and ∇: L2(Ω)→H−1(Ω)n are negative adjoints of each other.

From the proof of Theorem 13.9 we recall that there exists an open covering (Ωj)j=1,..,m of Ω, where each Ωj is star-shaped with respect to the points in a ball B(xj, rj)⊆ oh Using the compactness of the embedding H01(Ω),→L2(Ω), one can show that this inequality implies. 13.8) if Ω is bounded with Lipschitz boundary.

Interlude: The Bochner integral for Hilbert space valued functions

The completeness is proved in the same way as the completeness of the scalar value L2(a, b). Show that every absolutely convergent series is convergent.).

Vector valued Sobolev spaces

The next result shows that the representative is in fact even continuous with values ​​in the smaller space H. 14.1). In several arguments in the following proof, it is used without mentioning that L2(R)⊗V is dense in L2(R;V), and similarly for V∗.

Lions’ representation theorem

The non-autonomous equation

Here for existence we need the main equality (14.1) only for functions on W, for which it is in fact trivial. In contrast, uniqueness is more technical, since (14.1) is needed for a rather general class of functions.

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