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APPEARED IN BULLETIN OF THE

arXiv:math/9204239v1 [math.AP] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 294-298A SHARP POINTWISE BOUND FOR FUNCTIONSWITH L2-LAPLACIANS ON ARBITRARY DOMAINSAND ITS APPLICATIONSWENZHENG XIEAbstract. For all functions on an arbitrary open set Ω⊂R3 with zero bound-ary values, we prove the optimal boundsupΩ|u| ≤(2π)−1/2ZΩ|∇u|2 dxZΩ|∆u|2 dx1/4.The method of proof is elementary and admits generalizations.

The inequalityis applied to establish an existence theorem for the Burgers equation.1. IntroductionIn this note we announce the proof of the inequalitysupΩ|u| ≤1√2πZΩ|∇u|2 dxZΩ|∆u|2 dx1/4(1)for functions with zero boundary values on three-dimensional domains.

The domainΩcan be any open set and the constant 1/√2π is optimal. This best possible resultis obtained by a new and elementary method, which is apparently also applicableto other elliptic operators.

Thus, many known inequalities can be improved andnew ones derived. Some of these will be given by the author in separate papers.Such inequalities are used in the study of nonlinear differential equations, see [1]and [2].For smoothly bounded domains, one can combine the Sobolev inequality (see[3])supΩ|u| ≤C1(Ω)∥∇u∥1/2L2(Ω)∥u∥1/2H2(Ω)with the a priori estimate (see [4])∥u∥H2(Ω) ≤C2(Ω)∥∆u∥L2(Ω)(2)to obtainsupΩ|u| ≤C3(Ω)∥∇u∥1/2L2(Ω)∥∆u∥1/2L2(Ω),(3)Received by the editors January 20, 1991 and, in revised form, September 10, 1991.1991 Mathematics Subject Classification.

Primary 26D10, 35B45, 35Q20.Partially supported by NSERC (Canada). The contents of this paper have been presented tothe Annual Meeting of the American Mathematical Society, January 16–19, 1991.c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2WENZHENG XIEwhere Ci(Ω) are constants depending on the domain Ω. However, the elliptic esti-mate (2) fails to hold for domains with reentrant corners [5].It was suggested to the author by Professor J. G. Heywood that (3) should bevalid for nonsmooth domains as well, and its generalization to the Stokes operatorwould yield results for the Navier-Stokes equations in nonsmooth domains.

Here,in §4, we use (1) to derive a priori estimates and prove an existence theorem for theinitial-boundary value problem of the Burgers equation with H1 initial data, in anarbitrary open set, for the first time.2. The main resultsLet Ωbe an arbitrary open set in R3 .

Let ∥· ∥denote the L2(Ω) norm. Thehomogeneous Sobolev space bH10(Ω) is defined to be the completion of C∞0 (Ω) inthe Dirichlet norm ∥∇· ∥, where ∇is the gradient.

Let ∆denote the Laplacianin the sense of distributions. Our main result isTheorem 1.

For all u ∈bH10(Ω) with ∆u ∈L2(Ω) , there holdssupΩ|u| ≤1√2π ∥∇u∥1/2 ∥∆u∥1/2.The constant 1/√2π is optimal for each Ω.The space bH10(Ω) contains the standard Sobolev space H10(Ω) .It containsfunctions that are not square integrable for some unbounded domains.If u ∈H10(Ω) , then using ∥∇u∥2 = −RΩu∆u dx ≤∥u∥∥∆u∥, we obtainCorollary 1. If u ∈H10(Ω) and ∆u ∈L2(Ω) , then u also satisfiessupΩ|u| ≤1√2π∥u∥1/4 ∥∆u∥3/4.In particular, we obtain a pointwise bound for any normalized eigenfunction ofthe Laplacian, in terms of its corresponding eigenvalue.Corollary 2.

If u satisfies−∆u = λu ,u ∈H10(Ω) ,∥u∥= 1 ,thensupΩ|u| ≤λ3/4√2π .All of the above results are also valid for vector-valued or complex-valued func-tions. The constants in the corollaries, however, are not optimal.3.

Outline of proofThe proof of (1) has four steps.Step 1. First we assume that Ωis bounded, with a C∞boundary ∂Ω.

It is wellknown that there exist eigenfunctions {φn} of the Laplacian that form a completeorthonormal basis of L2(Ω) , satisfying−∆φn = λnφn ,φn|∂Ω= 0 ,where λn > 0 are the eigenvalues, n = 1, 2, . .

. .

L2-LAPLACIANS ON ARBITRARY DOMAINS3Let x0 ∈Ωand m ≥1 be fixed. For functions of the form u(x) = Pmn=1 cnφn(x) ,we haveu2(x0)∥∇u∥∥∆u∥=Xmn=1 cnφn(x0)2Xmn=1 λnc2n1/2 Xmn=1 λ2nc2n1/2 .This quotient is a smooth and homogeneous function of (c1, .

. .

, cm) in Rm\{0}.Hence, at some point (˜c1, . .

. , ˜cm) , it attains its maximum value.

The maximumvalue can be written as4√µmXn=1φn(x0)µ + λn2,where µ = Pmn=1 λ2n˜c2n/ Pmn=1 λn˜c2n.Step 2. We introduce the Green function G(x; x0, µ) for the Helmholtz equation∆G = µG −δ(x −x0) ,G|∂Ω= 0 .By the maximum principle, we have0 ≤G(x; x0, µ) ≤e−√µ|x−x0|4π|x −x0| ,∀x ∈Ω\{x0} ,the upper bound being the fundamental solution.

HenceZΩG2 dx ≤Z ∞0e−√µr4πr24πr2 dr =18π√µ.By Parseval’s equality and Green’s formula, we haveZΩG2 dx =∞Xn=1ZΩGφn dx2=∞Xn=1φn(x0)µ + λn2.Thereforeu2(x0)∥∇u∥∥∆u∥≤4√µmXn=1φn(x0)µ + λn2≤4√µZΩG2 dx ≤12π .Thus, (1) is true for any function of the form u(x) = Pmn=1 cnφn(x) .Step 3. Now, let u be any function in bH10(Ω) such that ∆u ∈L2(Ω) .

Let unbe the projection of u in span{φ1, . .

. , φn}.

We have ∥∇un∥≤∥∇u∥, ∥∆un∥≤∥∆u∥, and limn→∞un = u in L2(Ω) . It follows that (1) remains valid.Step 4.

Now we proceed to prove Theorem 1. We can choose a sequence ofbounded domains Ωn with smooth boundaries such that Ω1 ⊂Ω2 ⊂· · · andS∞n=1 Ωn = Ω.

For each n ≥1 , there exists a unique un ∈bH10(Ωn) such thatZΩn∇un · ∇v dx =ZΩn∇u · ∇v dx,∀v ∈bH10(Ωn) ,by the Riesz representation theorem. From this we obtain ∥∇un∥L2(Ωn) ≤∥∇u∥,∆un = ∆u|Ωn , and limn→∞un = u inbH10(Ω) , hence in L6(Ω) .

Therefore theproof of (1) is completed.

4WENZHENG XIELet u(x) = (1 −e−|x|)/|x|; then we havesupx∈R3 |u(x)| = 1 ,ZR3 |∇u|2 dx = 2π ,ZR3 |∆u|2 dx = 2π .Hence the equality in (1) holds for u. By cutting-offu, we explicitly construct asequence of functions un with compact support such thatun(0) →1 ,ZR3 |∇un|2 dx →2π ,ZR3 |∆un|2 dx →2π ,as n →∞.Given any open set Ω, by scaling, we obtain a new sequence offunctions with compact support in Ω.

Since the product∥∇un∥1/2L2(R3) ∥∆un∥1/2L2(R3)is scale invariant, it is seen that the constant 1/√2π in (1) is the best possible.4. Application to the Burgers equationThe time-dependent Burgers equation∂u∂t + u · ∇u = ν∆u(4)is sometimes studied for its analogy with the Navier-Stokes equations, with thethree-dimensional vector-valued function u(x, t) representing the velocity field andthe positive constant ν the viscosity coefficient.

We consider as spatial domain anarbitrary open set in R3 and seek u that vanishes on the boundary and takes aninitial value u0 ∈bH10(Ω)3. From (4) and the vector version of (1), we have12ddt ∥∇u∥2 + ν ∥∆u∥2=ZΩu · ∇u · ∆u dx≤sup |u| ∥∇u∥∥∆u∥≤1√2π ∥∇u∥3/2 ∥∆u∥3/2 .By using Young’s inequality and a comparison theorem, we obtain∥∇u(t)∥2 ≤∥∇u0∥2p1 −t/T,andZ t0∥∆u(s)∥2 ds ≤∥∇u0∥22ν1 −6pt/T q1 −pt/T,for 0 ≤t < T , whereT =256π2ν327∥∇u0∥4 .Beginning with these a priori estimates, and using the methods of [1] and [6], thefollowing theorem is established [7].

L2-LAPLACIANS ON ARBITRARY DOMAINS5Theorem 2. For any open Ω⊂R3 and any u0 ∈bH10(Ω)3, there exists a uniquefunctionu ∈C([0, T ), bH10(Ω))3 ∩C∞(Ω× (0, T ))3 ∩C∞((0, T ), L∞(Ω))3 ,satisfying the Burgers equation (4) and taking the initial value u0.AcknowledgmentsI am grateful to J. G. Heywood for suggesting the topic for this paper and forhis helpful advice.

I would also like to thank L. Rosen for helpful discussions.References1. J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem.

I. Regularity of solutions and second-order error estimates for spatial dis-cretization, SIAM J. Numer. Anal.

19 (1982), 275–311.2. R. Temam, Navier-Stokes equations and nonlinear functional analysis, SIAM, Philadelphia,PA, 1983.3.

R. A. Adams and J. J. Fournier, Cone conditions and properties of Sobolev spaces, J. Math.Anal. Appl.

61 (1977), 713–734.4. O.

A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math.Sci., vol.

49, Springer-Verlag, New York, 1985.5. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs Stud.

Math., vol. 24,Pitman Publishing Inc., Boston, MA, 1985.6.

J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solu-tions, Indiana Univ. Math.

J. 29 (1980), 639–681.7.

W. Xie, A sharp pointwise bound for the Poisson equation in arbitrary domains and itsapplications to Burgers’ equation, thesis, University of British Columbia, 1991.School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455E-mail address: xie@s5.math.umn.edu

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