We introduce a simple criterion to check coercivity of bilinear forms on subspaces of Hilbert-spaces. The presented criterion allows to derive many standard and non-standard variants of Poincar\'e- and Friedrichs-type inequalities with very little effort

Gräser, C.

English

Repository: Freie Universität Berlin (FU), Math Department (fu_mi_publications)

A NOTE ON POINCARE´- AND FRIEDRICHS-TYPEINEQUALITIESCARSTEN GRA¨SERAbstract. We introduce a simple criterion to check coercivity of bilinearforms on subspaces of Hilbert-spaces. The presented criterion allows to derivemany standard and non-standard variants of Poincare´- and Friedrichs-typeinequalities with very little effort.1. IntroductionPoincare´- and Friedrichs-type inequalities play an important role in existencetheory for elliptic and parabolic partial differential equations because they allowto show coercivity of bilinear forms on subspaces of Sobolev spaces. In manyapplications those bilinear forms are obtained from the natural inner product ofthe Sobolev space by incorporating non-constant coefficients, dropping lower orderderivatives, or adding modified lower order terms. The considered subspaces areobtained by imposing boundary or other conditions on solutions. As a consequencea variety of Poincare´- and Friedrichs-type inequalities where proposed to deal withdifferent bilinear forms or constraining conditions. Each of these is often provedindependently.The aim of this paper is not to show a specific new variant of such an inequality.Instead we give a simple criterion to check coercivity of bilinear forms, which allowsto link many variants of Poincare´- and Friedrichs-type inequalities. The purpose ofthis is two-fold: On the one hand it allows to avoid time consuming research for apublished suitable variant in non-standard situations. On the other hand in can beused in teaching to easily derive the most common variants with little effort.The main criterion is introduced in Section 2 in a general Hilbert-space setting.In Section 3 we show how many variants of Poincare´- and Friedrichs-type inequal-ities can be derived from a single one using this criterion. Examples incorporatethe most common, as well as some non-standard variants. Finally, we apply this inSection 4 to derive coercivity for special boundary conditions of forth- and eighthorder problems.2. Coercivity on subspaces of Hilbert-spacesIn the following we will call a bilinear form a( · , ·) coercive on a normed spaceV with constant γ > 0 ifγ‖v‖2 ≤ a(v, v) ∀v ∈ V.First we show an auxiliary result linking the angle between subspaces to norms oforthogonal projections.Key words and phrases. Poincare inequality, coercivity, plate problem.1arXiv:1512.02842v1  [math.AP]  9 Dec 20152 GRA¨SERLemma 1. Let H be a Hilbert-space, V,W ⊂ H closed subspaces of H withdim(W ) <∞ and V ∩W = {0}. Then we have0 ≤ supv∈V \{0},w∈W\{0}(v, w)‖v‖‖w‖ = α(V,W ) < 1,(1)i.e., ](V,W ) = acos(α(V,W )) > 0.Proof. Assume that this is not the case, then there are sequences vn ∈ V andwn ∈ W with ‖vn‖ = ‖wn‖ = 1 for all n and (vn, wn) → 1. By compactness thereare subsequences (wlog. also denoted by vn and wn) and v ∈ V , w ∈ W withvn ⇀ v and wn → w. Then we have(vn, wn)→ (v, w) = 1,‖w‖ = 1, and furthermore by Hahn–Banachs theorem ‖v‖ ≤ 1. As a consequencewe get ‖v − w‖2 = ‖v‖2 − 2(v, w) + ‖w‖2 ≤ 0 and thus 0 6= v = w ∈ V ∩W whichcontradicts the assumption. Lemma 2. Let H be a Hilbert-space, V,W ⊂ H closed subspaces of H withdim(W ) <∞ and V ∩W = {0}. Then the orthogonal projections P : H →W and(I − P ) : H →W⊥ satisfy the inequalities‖Pv‖ ≤ α(V,W )‖v‖, ‖v‖ ≤ 1β(V,W )‖(I − P )v‖ ∀v ∈ Vwith constants α(V,W ) = cos(](V,W )) < 1 and β(V,W ) = sin(](V,W )) > 0.Proof. By Lemma 1 we have α(V,W ) = cos(](V,W )) < 1 and thus β(V,W ) =√1− α(V,W )2 > 0. Now let v ∈ V . Then we have‖Pv‖2 = (Pv, v − (I − P )v) = (Pv, v) ≤ α(V,W )‖Pv‖‖v‖and thus ‖Pv‖ ≤ α(V,W )‖v‖. Hence we get‖v‖2 = ‖(I − P )v‖2 + ‖Pv‖2 ≤ ‖(I − P )v‖2 + α(V,W )2‖v‖2.Subtracting α(V,W )2‖v‖2 and taking the square root provides the assertion. Using these results we are now ready to show a general criterion for coercivityon subspaces.Proposition 1. Let H be a Hilbert-space and a( · , ·) : H ×H → R a continuoussymmetric bilinear form with finite dimensional kernel ker a = {v ∈ H | a(v, v) =0}. Furthermore assume that a( · , ·) is coercive on (ker a)⊥ with constant γ > 0.Then a( · , ·) is coercive with constant γβ(V, ker a)2 > 0 on any closed subspace Vof H with V ∩ ker a = {0}.Proof. Let v ∈ V and P : H → ker a the orthogonal projection into ker a. ThenLemma 2 and coercivity on (ker a)⊥ = (I − P )(H) provide γβ(V, ker a)2 > 0 andγβ(V, ker a)2‖v‖2 ≤ γ‖(I − P )v‖2 ≤ a((I − P )v, (I − P )v) = a(v, v).In many situation coercivity is not obtained by restriction to suitable subspaces,but by augmenting the bilinear form a( · , ·) in order to obtain coercivity on thewhole space.A NOTE ON POINCARE´- AND FRIEDRICHS-TYPE INEQUALITIES 3Proposition 2. In addition to the assumptions of Proposition 1 assume thatb( · , ·) : H × H → R is a continuous symmetric bilinear form which is positivesemi-definite on H and positive definite on ker a. Then a( · , ·) + b( · , ·) is coerciveon H.Proof. Since ker a is finite dimensional, positive definiteness implies coercivity ofb( · , ·) on ker a. Hence we have for v1 = Pv and v2 = v − Pv‖v‖2 = ‖v − Pv‖2 + ‖Pv‖2 ≤ ‖v − Pv‖2 + C2b(Pv, Pv)≤ ‖v − Pv‖2 + 2C2(b(v, v) + b(v − Pv, v − Pv))≤ (1 + C3)‖v − Pv‖2 + 2C2b(v, v)≤ C4a(v, v) + 2C2b(v, v).3. Poincare´ inequalities in Hm(Ω)Now we consider Poincare´ type inequalities in Hm(Ω) with m ∈ N0. Throughoutthis section let Ω ⊂ Rd be a bounded domain with Lipschitz boundary. On Hm(Ω)we use the inner product(u, v)m =∑|s|≤m∫ΩDsuDsv dxand the induced norm ‖·‖m, where we used the classical multi-index notation.In the following we investigate coercivity of the bilinear form〈u, v〉m =∑|s|=m∫ΩDsuDsv dxand augmented variants on subspaces of Hm(Ω). Note that this bilinear forminduces the Hm-seminorm | · |m = 〈 · , · 〉1/2m . The main ingredients are the charac-terization of the kernel of 〈 · , · 〉m and the coercivity on its orthogonal complement.Lemma 3. The kernel of 〈 · , · 〉m is given by ker(〈 · , · 〉m) = Pm−1 where Pk is thespace of polynomials with degree ≤ k on Rd.Proof. Let 〈v, v〉m = 0. Then we have Dsv = 0 for all multi-indices s with |s| = mand hence v ∈ Pm−1. Next we show that 〈 · , · 〉m is coercive on the orthogonal complement of Pm−1.To this end we need the following classical version of the Poincare´ inequality onHm(Ω). All other versions will be derived from this one.Theorem 1. There is a constant C > 0 such that‖v‖2m ≤ C(|v|2m +∑|s|<m(∫ΩDsv dx)2) ∀v ∈ Hm(Ω).Proof. See [1, Theorem 7.2]. Lemma 4. Let P : H1(Ω)→ Pm−1 be the orthogonal projection into Pm−1. Then‖v − Pv‖2m ≤ C|v|2m ∀v ∈ Hm(Ω)for the same constant C as in Theorem 1.4 GRA¨SERProof. We will use the modified inner product〈〈〈u, v〉〉〉m =∑|s|=m∫ΩDsuDsv dx+∑|s|<m∫ΩDsu dx∫ΩDsv dxon Hm(Ω). By Theorem 1 this induces an equivalent norm ||| · |||m. The orthogonalprojection into Pm−1 with respect to ||| · |||m will be denoted by Pˆ .Now let u ∈ V . Utilizing Dsv = 0 for |s| = m and any v ∈ Pm−1 and Galerkin-orthogonality we get0 = 〈〈〈u− Pˆ u, v〉〉〉m =∑|s|<m∫ΩDs(u− Pˆ u) dx∫ΩDsv dx ∀v ∈ Pm−1.We will inductively show ∫ΩDs(u− Pˆ u) dx(2)for all |s| < m. For s = (0, . . . , 0) this follows from testing with v = xs = 1 = Dsv.Now let |s′| < m, assume that (2) is true for all |s| < |s′|, and set v = xs′ . Thenwe have Drv = 0 for all |r| > |s′| and |r| = |s| with r 6= s. Hence testing with vgives (2) with s = s′.As a consequence of Theorem 1, identity (2), and DsPˆ v = 0 for |s| = m we get‖u− Pu‖2m ≤ ‖u− Pˆ u‖2m ≤ C|||u− Pˆ u|||2m = C|u− Pˆ u|2m = C|u|2m.(3)As an immediate consequence of this and (Pm−1)⊥ = (I − P )(Hm(Ω)) we get:Corollary 1. The bilinear form 〈 · , · 〉m is coercive on (ker(〈 · , · 〉m))⊥ = (Pm−1)⊥.As a consequence of the kernel characterization in Lemma 3 and the coercivityresult in Corollary 1 we can use Proposition 1 to show coercivity on subspaces ofHm(Ω).Corollary 2. Let V ⊂ Hm(Ω) be a closed subspace with V ∩ Pm−1 = {0}. Then〈 · , · 〉m is coercive on V and | · |m is equivalent to ‖·‖m on V .Now we will show some examples of Poincare´- or Friedrichs-type inequalities orrelated coercivity results.Example 1. Then there is a constant Cp such that‖v − 1|Ω|∫Ωv dx‖21 ≤ Cp|v|21 ∀v ∈ H1(Ω).Proof. Since v 7→ 1|Ω|∫Ωv dx ∈ P0 is an orthogonal projection this is a special caseof Lemma 4. Example 2. Let Γ ⊂ ∂Ω with nonzero measure. Then 〈 · , · 〉1 is coercive onH1Γ,∫,0(Ω) = {v ∈ H1(Ω) | ∫Γv ds = 0}.Proof. By the trace theorem v 7→ v|Γ ⊂ L2(Γ) is a continuous map and henceH1Γ,∫,0(Ω) ⊂ H1(Ω) is a closed subspace. Since H1Γ,∫,0(Ω) ∩ P0 = {0} Corollary 2provides the assertion. Example 3. Let Γ ⊂ ∂Ω with nonzero measure. Then 〈 · , · 〉1 is coercive onH1Γ,0(Ω) = {v ∈ H1(Ω) | v|Γ = 0}.A NOTE ON POINCARE´- AND FRIEDRICHS-TYPE INEQUALITIES 5Proof. We only need to note that H1Γ,0(Ω) is a closed subspace of H1Γ,∫,0(Ω). Example 4. Let Γ ⊂ ∂Ω with nonzero measure. Then‖v‖21 ≤ C|v|21 +|Ω||Γ|2(∫Γv ds)2 ∀v ∈ H1(Ω)where C is the coercivity constant from Example 1.Proof. Let v ∈ H1(Ω) then Example 1 provides‖v‖21 ≤ ‖v − Pv‖21 + ‖Pv‖21 ≤ C|v|21 + ‖Pv‖20 = C|v|21 +|Ω||Γ|2(∫Γv ds)2.As a direct consequence we get a version of Friedrichs’ inequality with boundaryintegrals.Example 5. Let Γ ⊂ ∂Ω with nonzero measure. Then there is a constant C with‖v‖21 ≤ C(|v|21 + ‖v‖2L2(Γ))∀v ∈ H1(Ω).Example 6. Let d = 1, 2, 3 and p1, . . . , pd+1 ⊂ Ω affine independent. Then 〈 · , · 〉2is coercive on V = {v ∈ H2(Ω) | v(p1) = · · · = v(pd+1) = 0}.Proof. By the Sobolev embedding V is closed. Furthermore V ∩ P1 = {0} andCorollary 2 provides the assertion. Example 7. The bilinear form 〈 · , · 〉2 is coercive on H2(Ω) ∩H10 (Ω).Proof. Since Ω is bounded we have H10 (Ω)∩P1 = {0}. Hence Corollary 2 providesthe assertion. Example 8. Let d = 1, 2, 3 and p1, . . . , pd+1 ⊂ Ω affine independent. Then thebilinear form 〈 · , · 〉2 + b( · , ·) withb(u, v) =d+1∑i=1u(pi)v(pi)is coercive on H2(Ω).Proof. Symmetry and positive semi-definiteness of b( · , ·) are obvious. Positivedefiniteness on P1 follows from affine independence. Finally, the Sobolev embeddingimplies continuity such that Proposition 2 provides the assertion. 4. Coercivity of the bi- and quadruple-Laplacian operatorIn the following we show coercivity of the operators ∆2 and ∆4 with variousboundary conditions. Since such operators often arise in the context of plate-likeproblems, we restrict our considerations to piecewise smooth domains Ω ⊂ R2.In the following ν and τ will denote piecewise smooth oriented unit normal andtangential fields.We are especially interested in periodic boundary conditions. To this end wedefine for the special case of a rectangle Ω the periodic spacesC∞p (Ω) = {v|Ω | v ∈ C∞(R2) is Ω-periodic}, Hkp (Ω) = C∞p (Ω)‖·‖k.Lemma 5. Let V be any of the spaces6 GRA¨SER• H20 (Ω),• H2(Ω) ∩H10 (Ω),• H2p (Ω) with rectangular Ω,then ‖∆v‖20 = |v|22 for all v ∈ V .Proof. Let V˜ a corresponding dense subspace of smooth functions given by C∞0 (Ω),C∞(R2)∩H10 (Ω), or C∞p (Ω), respectively. Then partial integration for v ∈ V˜ gives‖∆v‖20 = |v|22 +2∑i,j=1∫∂Ω∂iv∂jjvνi − ∂jiv∂ivνj ds= |v|22 +∫∂Ω∂v∂ν∆v −∇ ∂∂ν v · ∇v ds= |v|22 +∫∂∂v∂ν∂2∂τ2 v − ∂∂τ ∂∂ν v ∂∂τ v ds.(4)For the case V˜ = C∞0 (Ω) the boundary term obviously vanishes. For V =H2(Ω) ∩ H10 (Ω) we have v|∂Ω = 0 and thus the boundary term is zero. For V =H2p (Ω) we can split the boundary according to ∂Ω = ΓW ∪ΓE ∪ΓN ∪ΓS such thatwe have (up to translation)v|ΓW = v|ΓE , v|ΓN = v|ΓS , ∂∂ν v|ΓW = − ∂∂ν v|ΓE , ∂∂ν v|ΓN = − ∂∂ν v|ΓS .Now the minus sign (resulting from the flipped orientation of the normal) impliesthat boundary integrals from opposing boundary segments cancel out. Proposition 3. Let V be any of the spaces• H20 (Ω),• H2(Ω) ∩H10 (Ω),• {v ∈ H2p (Ω) |∫∂Ωv ds = 0} with rectangular Ω,• {v ∈ H2p (Ω) |∫Ωv dx = 0} with rectangular Ω,then the bilinear form a(u, v) =∫Ω∆u∆v dx is coercive on V .Proof. By Lemma 5 we have a(v, v) = |v|22 for v ∈ V . Since P1 ∩ V = {0} for anychoice of V , Corollary 2 now provides the assertion. For the quadruple-Laplacian we get similar results:Lemma 6. Let V be any of the spaces• H40 (Ω),• H4(Ω) ∩H30 (Ω),• H4∆(Ω) = {v ∈ H4(Ω) | v = 0, ∆v = 0 on ∂Ω} with rectangular Ω• H4p (Ω) with rectangular Ω,then ‖∆2v‖20 = |v|24 for all v ∈ V .Before giving the proof we note that v ∈ H4(Ω) ∩H30 (Ω) implies the boundaryconditionsv = ∂∂ν v =∂2∂ν2 v = 0,(5)and hence also∂∂τ v =∂2∂τ2 v = ∆v = 0(6)A NOTE ON POINCARE´- AND FRIEDRICHS-TYPE INEQUALITIES 7on ∂Ω. Thus we have H4(Ω) ∩H30 (Ω) ⊂ H4∆(Ω). Conversely the boundary condi-tions for v ∈ H4∆(Ω) imply∂∂τ v =∂2∂τ2 v = 0,∂2∂ν2 v = ∆v − ∂2∂τ2 v = 0.(7)Proof. Let V˜ a corresponding dense subspaces of smooth functions given by C∞0 (Ω),C∞(R2) ∩H30 (Ω), C∞(R2) ∩H4∆(Ω), or C∞p (Ω), respectively.Now let v ∈ V˜ and w = ∆v ∈ H2(Ω). In view of (5)–(7) we can apply Lemma 5to get‖∆2v‖20 = |∆v|22 =∑|s|=2‖Ds∆v‖20 =∑|s|=2‖∆Dsv‖20.Hence it remains to show ‖∆z‖20 = |z|22 for z = Dsv and |s| = 2. To this end weagain apply partial integration as in (4) to get‖∆z‖20 = |z|22 +∫∂Ω∂z∂ν∂2∂τ2 z − ∂∂τ ∂∂ν z ∂∂τ z ds.(8)By local coordinate transformation we find that there are piecewise smooth func-tions α, β, γ, independent of v, such thatz = Dsv = α ∂2v∂ν2 + β∂2v∂τ2 + γ∂2v∂τ∂ν a.e. on ∂Ω.(9)For V = H4(Ω) ∩ H30 (Ω) the boundary term vanishes because the boundary con-ditions (5) and (6) for v reduce (9) to z|∂Ω = 0. For V = H4∆(Ω) the boundaryconditions (7) reduce (9) toz = γ ∂2v∂τ∂ν a.e. on ∂Ω.Noting that γ = const for rectangular Ω, the boundary term again vanishes in thiscase. Finally we note that for V = H4p (Ω) periodicity of v implies periodicity of z,such that the boundary term vanishes by the same arguments as in Lemma 5. Proposition 4. Let V be any of the spaces• H40 (Ω),• H4(Ω) ∩H30 (Ω),• H4∆(Ω) = {v ∈ H4(Ω) | v = 0, ∆v = 0 on ∂Ω} with rectangular Ω• {v ∈ H4p (Ω) |∫∂Ωv ds = 0} with rectangular Ω,• {v ∈ H4p (Ω) |∫Ωv dx = 0} with rectangular Ω,then the bilinear form a(u, v) =∫Ω∆2u∆2v dx is coercive on V .Proof. By Lemma 6 we have a(v, v) = |v|24 for v ∈ V . In view of Corollary 2 itremains to show P3 ∩ V = {0} for all choices of V . To this end let p ∈ P3 ∩ V .For V = H4(Ω)∩H3(Ω) the boundary conditions (5) and (6) provide for |s| ≤ 2that Dsp = 0 on ∂Ω. For |s| = 2 we have Dsp ∈ P1 and thus Dsp = 0 on Ω.Hence p is bilinear and we get for |s| = 1 that Dsp ∈ P1 and thus Dsp = 0. As aconsequence p is constant which gives p = 0. Since H40 (Ω) ⊂ H4(Ω) ∩ H30 (Ω) wehave also covered this case.For V = H4∆(Ω) with rectangular Ω the boundary conditions (7) imply for s =(2, 0) and s = (0, 2) that Dsp = 0 on ∂Ω which, together with Dsp ∈ P1 impliesDsp = 0 on Ω. Hence p is bilinear on the rectangle Ω which together with p|∂Ω = 0gives p = 0.8 GRA¨SERFinally periodicity of p ∈ H4p (Ω) implies that p = const which together with∫∂Ωp ds = 0 or∫Ωp dx = 0 gives p = 0. References[1] J. Wloka. Partial Differential Equations. Cambridge Univiversity Press, Cambridge, 1987.Carsten Gra¨ser, Freie Universita¨t Berlin, Institut fu¨r Mathematik, Arnimallee 6,D - 14195 Berlin, GermanyE-mail address: graeser@math.fu-berlin.de

A note on Poincaré- and Friedrichs-type inequalities

http://publications.imp.fu-berlin.de/2007/1/1512.02842v1.pdf

A note on Poincaré- and Friedrichs-type inequalities

Abstract

Similar works

Full text

Available Versions

Repository: Freie Universität Berlin (FU), Math Department (fu_mi_publications)