A Product Property of Sobolev Spaces with Application to Elliptic Estimates

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1 Rend. Sem. Mat. Univ. Padova Manoscritto in corso di stampa pervenuto il 23 luglio 2012 accettato l 1 ottobre 2012 A Product Property of Sobolev Spaces with Application to Elliptic Estimates by Henry C. Simpson Department of Mathematics University of Tennessee Knoxville TN USA and Scott J. Spector Department of Mathematics Southern Illinois University Carbondale IL USA November Abstract. In this paper a Sobolev inequality which generalizes the ordinary Banach algebra property of such spaces is established; for p [1 ) n m Z and m 2 that satisfy m > n/p φψ mp K sup φ ψ mp ( ψ m 1q ψ m 1p φ mp for all φ ψ W mp () that satisfy spt ψ and domains R n that are nonempty open and satisfy the cone condition. Here q = p if p > n q (n/υ /(n p)] if n > p q (n/υ ) if p = n K = K(n p m q C) where C is the cone from the cone condition and Υ := [n/p] the largest integer less than or equal to n/p. Mathematics Subject Classifications (2010): 46E35 35J57 74B15 Key words: Elasticity elliptic regularity Sobolev estimate systems of partial differential equations. 1 Introduction; Sobolev Spaces A standard classical methodology used to obtain a priori estimates for elliptic systems of partial differential equations is to first prove the required estimate when the system has 1

2 2 H. C. Simpson and S. J. Spector constant coefficients and the region has smooth boundary and then use a partition of unity to extend the estimate to coefficients that depend on position and regions that are less regular. For example Agmon Douglis and Nirenberg [3 4] first establish the estimate (in the notation from Elasticity): for all u C (; R n ) that satisfy u = 0 in a neighborhood of D := \S [ u m1p N Div C[ u] m 1p C[ u]n ] m 1 p ps u p (1.1) where C : M n n M n n is a constant linear mapping of the n n matrices M n n and is a ball with S = or is a half-ball and S is the flat portion of the boundary of. Here m Z p (1 ) n is the outward unit normal to the boundary u p p := u p mp := α m u(x) p dx ( u) i := u x i D α u p p (Div M) i := n j=1 M ij x j and α = (α 1... α n ) is a multi-index with α = α 1 α n and D α = α 1 x 1... αn x n. For a general bounded open region R n a suitable open covering of and respectively by balls and half-balls together with a partition of unity can then be used (see [3]) to prove (1.1) for C(x) : M n n M n n whose components are m-times continuously differentiable on. More generally if one wants to establish 1 (1.1) for C W mp () then one can make use of Moser s [7 pp ] tame inequality (see Klainerman and Majda [6 pp ] for a nice proof): If 1 p < and k Z then there exists a constant C = C(n p k) > 0 such that for all φ ψ W kp (R n ) L (R n ). C 1 φψ kpr n φ R n ψ kpr n ψ R n φ kpr n (1.2) However (1.2) is an inequality for Sobolev functions defined on all of R n while in practice one must make use of this inequality for Sobolev functions defined on a bounded open region R n. This presents no difficulties when the boundary of is sufficiently smooth since one can then use standard extension results to obtain a version of (1.2) for such domains. When the boundary of the region is not smooth there are some unresolved difficulties. 2 1 See e.g. [12] for a proof of (1.1) when C is a Sobolev function. See e.g. Ragusa [11] and the references therein for interior regularity when C is discontinuous. 2 See Maz ya and Shaposhnikova [8 Chapter 7] for regions whose boundary can be parametrized by an appropriate Sobolev function. See also Nečas [10].

3 A Product Property of Sobolev Spaces 3 The main purpose of this paper is to provide a partial resolution of these difficulties by proving an inequality which is similar to (1.2) and which is also useful for elliptic estimates for regions that satisfy (only) a cone condition. In particular we show that if R n is nonempty open and satisfies the cone condition and if p [1 ) n m Z m 2 and p (n/m n) then for any q (n/[ n p ] /(n p)] there is a constant K = K(n p m q C) such that φ 1 φ 2 mp K sup φ 1 φ 2 mp ( φ 2 m 1q φ 2 m 1p φ 1 mp (1.3) for all φ 1 φ 2 W mp () that satisfy spt φ 2. Here C is the cone from the cone condition 3 and [[x] is the largest integer less than or equal to x. We note that our inequality unlike (1.2) has the interesting feature that its dependence on the supremum of the first function is limited to the region that supports the second function. Our initial motivation for studying such inequalities originated in problems of global bifurcation 4 for the strongly-elliptic system that governs the equilibrium of elastic materials. In this context inequality (1.3) extends results of Valent [13 pp ] that are used to improve estimates in [3 4] in order to apply them to elasticity. Our proof makes use of the following special cases of the usual Sobolev inequalities. Proposition (See e.g. [2 pp ]). Let R n n 1 be a nonempty open region that satisfies the cone condition. Suppose 1 p < k Z and j N. Define p k := /(n kp) if n > kp and p k := otherwise. Then there exist a constant K = K(n p k j q C) where C is the cone from the cone condition that has the following properties. I. (Sobolev Imbedding Theorem) If k > n/p then W kp () C B () with sup φ K φ kp for all φ W kp (). II. (Gagliardo-Nirenberg-Sobolev Inequality) If k n/p then W kjp () W jq () with φ jq K φ kjp for all φ W kjp () and q [p p k ] if p k < and q [p ) otherwise. Here C B () := {φ C() : φ L ()} which is a Banach space under the L -norm. 3 That is every x is the vertex of cone that is contained in and is a rigid deformation of C. See e.g. [1 p. 66] or [2 p. 82]. 4 See e.g. [5 9].

4 4 H. C. Simpson and S. J. Spector 2 The Product Property For a Sobolev function ψ W kp () k N p [1 ) we define the support of ψ by spt ψ := \ {x : ψ(z) = 0 for a.e. z in some open neighborhood of x}. Thus since the complement of spt ψ is an open set D α ψ = 0 a.e. on \(spt ψ) for α k. Consequently if φ ψ W kp () k > n/p and spt ψ then φ(d γ ψ) L p () for γ k and φ(d γ ψ) ( ) p sup φ D γ ψ. (2.1) p The main result of this paper is the following theorem which generalizes the usual Banach algebra property of W mp m > n/p. See Valent [13 pp ] for similar results. Theorem. Let R n n Z be a nonempty open region that satisfies the cone condition. Suppose 1 p < m Z with m > n/p and is measurable. Then for every q ( n Υ np/(n p)] if n > p and for every q ( n Υ ) if p = n there exists a constant K = K(n p m q C) > 0 where C is the cone from the cone condition for and Υ := [ n p ] such that the following are satisfied. (a) If m = 1 then for all φ ψ W 1p () that satisfy spt ψ ( φψ 1p 2 sup φ ψ 1p sup ψ φ 1p. (2.2) (b) If m 2 and n p then for all φ ψ W mp () that satisfy spt ψ φψ mp K sup φ ψ mp ( ψ m 1q ψ m 1p φ mp. (2.3) (c) If m 2 and p > n then for all φ ψ W mp () that satisfy spt ψ [( φψ mp K sup φ ψ mp ψ m 1p φ mp. (2.4) Remark. Note that the constant K is independent of. When has finite volume one can combine the term ψ m 1p with its upper bound ψ m 1q in (2.3) however the constant K will then depend on the volume of. Proof of (2.2). To simplify the notation we drop the but leave the on the appropriate norms. To prove (2.2) we first note that since p > n without loss of generality we may

5 A Product Property of Sobolev Spaces 5 assume by the Sobolev imbedding theorem that φ ψ W 1p () C B (). Next φψ 1p φψ 0p (φψ) 0p and in view of (2.1) ( ) φψ 0p sup φ ψ 1p. However (φψ) = φ ψ ψ φ so that with the aid of (2.1) which proves (2.2). (φψ) 0p φ ψ 0p ψ φ 0p ( ) ( ) sup φ ψ 1p sup ψ φ 1p Proof of (2.3) when [ n n p ] = m 1 and m 2. Note that n p. Let q ( m 1 np n p ] if n > p and q ( n m 1 ) if n = p. To prove (2.3) when [ n p ] = m 1 we first note for future reference that q > p q(m 1) > n (2.5) and if m > 2 then n > p and np n p n m 2. (2.6) Now let φ ψ W mp (). Then since mp > n φ ψ C B () by the Sobolev imbedding theorem while q [p p 1 ] (or q [p )) yields ψ W m 1q () by the Gagliardo-Nirenberg- Sobolev inequality. Next φψ mp D α (φψ) 0p α m = α m K α m β γ = α β γ = α where K := max c βγ only depends on n and m. c βγ (D β φ)(d γ ψ) 0p (D β φ)(d γ ψ) (2.7) 0p If β = 0 then γ m and hence by (2.1) φ(d γ ψ) 0p φ s D γ ψ 0p φ s ψ mp. (2.8) If β = m then γ = 0 and hence by the Sobolev imbedding theorem and (2.5) 2 (D β φ)ψ D β φ ψ K φ mp ψ m 1q. (2.9) 0p 0p

6 6 H. C. Simpson and S. J. Spector If γ = m 1 (and β 0) then β = 1. Define q so that 1 q 1 q = 1 p and p m 1 := /(n (m 1)p) if n > (m 1)p and p m 1 := otherwise. Then by (2.5) 2 q (p p m 1 ) and hence by Hölder s inequality and the Gagliardo-Nirenberg-Sobolev inequality (k = m 1) (D β φ)(d γ ψ) D β φ 0p 0q D γ ψ φ 1q ψ m 1q K φ mp ψ m 1q. (2.10) 0q Finally if 2 β m 1 then γ m 2. Note β γ m m 3 and n p (since n/p m 1 2). Thus by Hölder s inequality (D β φ)(d γ ψ) 0p D β φ 0 n (m β )p φ β n (m β )p D γ ψ 0 (m β )p ψ m β (m β )p. (2.11) Then in view of the Gagliardo-Nirenberg-Sobolev inequality (k = m β and k = β 1) φ β n (m β )p K φ mp ψ m β n m β K ψ m 1q (2.12) since q ( n m 1 n m 2 ] by (2.6) and the definition of q. The desired result (2.3) now follows in the case when [[ n p ] = m 1 from (2.7) (2.12). Proof of (2.3) when 1 [ n p ] < m 1. We prove (2.3) by induction on m. Note that n p. Define m := Υ 1 = [ n p ] 1. Then m > n/p m 1 and m 2. The induction starts at m = m. Then as we have just proven (2.3) is valid for m = m and any q in the appropriate interval. To continue the induction we assume (2.3) is valid for some m m and q and show it is valid for m 1 and the same q. Let φ ψ W m1p (). Then q ( n Υ np n p ] if n > p and q ( n Υ ) if n = p; consequently ψ W mq () by the Gagliardo-Nirenberg-Sobolev inequality. We again note (φψ) = ψ φ φ ψ and hence φψ m1p φψ mp (φψ) mp φψ mp ψ φ mp φ ψ mp. (2.13) However by the induction hypothesis φψ mp K sup φ ψ mp ( ψ m 1q ψ m 1p φ mp K sup φ ψ m1p ( ψ mq ψ mp φ m1p (2.14)

7 A Product Property of Sobolev Spaces 7 and similarly φ ψ mp K sup φ ψ m1p ( ψ mq ψ mp φ m1p (2.15) ψ φ mp K sup φ ψ mp ( ψ mq ψ mp φ m1p (2.16) since while by the Sobolev imbedding theorem (k = m) sup φ K φ m1p. (2.17) Inequality (2.3) with m replaced by m 1 now follows from (2.13) (2.17). Proof of (2.4). Note that p > n. First we prove (2.4) for m = 2. Let φ ψ W 2p (). Then (2.13) with m = 1 is valid however (2.14) (2.16) must be replaced by (see (2.2)) ( φψ 1p 2 sup φ ψ 1p sup ψ φ 1p (2.18) ( ψ φ 1p 2 sup φ ψ 1p sup ψ φ 1p and an appropriate estimate for φ ψ 1p. Clearly 1p 2p and since p > n the Sobolev imbedding theorem (k = 1) yields sup ψ K ψ 1p sup φ K φ 1p K φ 2p. (2.19) Thus we need only estimate the term φ ψ 1p which replaces (2.15) to finish the proof when m = 2. We note (φ ψ) = φ ψ ψ φ and hence φ ψ 1p φ ψ 0p (φ ψ) 0p φ ψ 0p φ ψ 0p ψ φ 0p. (2.20) In view of (2.1) the first two terms on the right-hand side of (2.20) satisfy ( ) ( ) φ ψ 0p sup φ ψ 2p φ ψ 0p sup φ ψ 2p (2.21) while the last term on the right-hand side of (2.20) satisfies ( ) ψ φ 0p sup φ ψ 1p. (2.22)

8 8 H. C. Simpson and S. J. Spector Inequality (2.4) with m = 2 now follows from (2.13) with m = 1 and (2.18) (2.22). To complete the proof we note that (2.4) can be obtained by induction on m for m > 2. However the required steps are identical to those in the proof of (2.3) when 1 [ n p ] < m 1 with the terms ψ mq and ψ m 1q deleted. Remark. If n p 1 m m = Υ1 := [ n p ]1 2 and is open then one can show that for all φ ψ W mp () that satisfy spt ψ and for every q ( n Υ np/(n p)] if n > p and for every q ( n Υ ) if p = n there exists a constant K = K(n p m q C) > 0 such that ( m m ) [ ] φψ mp K φ C k B ( ψ s) m kp ψ m 1q φ mp (2.23) where k=0 φ C k B () := α k sup D α φ(x). x Inequality (2.23) is obtained by induction on m. The initial step is the above proof of (2.3) when [ n p ] = m 1. The induction is then similar to that presented above in the proof of (2.3) when 1 [ n p ] < m 1. The only significant difference is that one does not use the estimate (2.17) but instead leaves the appropriate version of (2.16) as it is since each step in the induction argument will now add an additional derivative to the φ term that multiplies ψ m kp. Acknowledgement. The work of SJS was partially supported by the National Science Foundation under Grant No. DMS References [1] R. A. Adams Sobolev Spaces Academic Press [2] R. A. Adams and J. J. F. Fournier Sobolev Spaces 2 nd edition Academic Press [3] S. Agmon A. Douglis and L. Nirenberg Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I Comm. Pure Appl. Math. 12 (1959) [4] S. Agmon A. Douglis and L. Nirenberg Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II Comm. Pure Appl. Math. 17 (1964) [5] T. J. Healey and H. C. Simpson Global continuation in nonlinear elasticity Arch. Rational Mech. Anal. 143 (1998) 1 28.

9 A Product Property of Sobolev Spaces 9 [6] S. Klainerman and A. Majda Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids Comm. Pure Appl. Math. 34 (1981) [7] J. Moser A rapidly convergent iteration method and non-linear partial differential equations. I. Ann. Scuola Norm. Sup. Pisa (3) 20 (1966) [8] V. G. Maz ya and T. O. Shaposhnikova Theory of Multipliers in Spaces of Differentiable Functions Pitman [9] J. Shi and X. Wang On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differential Equations 246 (2009) [10] J. Nečas Les Méthodes Directes en Théorie des Équations Elliptiques Masson et Cie Éditeurs Paris [11] M. Ragusa Continuity of the derivatives of solutions related to elliptic systems Proc. Royal Soc. Edinburgh 136A (2006) [12] H. C. Simpson and S. J. Spector Applications of estimates near the boundary to regularity of solutions in linearized elasticity SIAM J. Math. Anal. 41 (2009) [13] T. Valent Boundary Value Problems of Finite Elasticity Springer 1988.