A three line proof for traces of H1 functions on special Lipschitz domains

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A three line proof for traces of H1 functions on special

Lipschitz domains

Sylvie Monniaux

To cite this version:

Sylvie Monniaux. A three line proof for traces of H1 functions on special Lipschitz domains. Ulmer

Seminare, 2014, pp.355-356. �hal-00984231�

A three lines proof for traces of H

1

functions

on special Lipschitz domains

Sylvie Monniaux

Aix-Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 7373

13453 Marseille, France

1

Introduction

It is well known (see [2, Theorem 1.2]) that for a bounded Lipschitz domain Ω ⊂ Rn_{, the trace operator}

Tr|∂Ω : C (Ω) → C (∂Ω) restricted to C (Ω) ∩ H

1_{(Ω) extends to a bounded operator from H}1_{(Ω) to L}2_(∂Ω)

and the following estimate holds:

kTr|_∂ΩukL2(∂Ω)≤ C kukL2(Ω)+ k∇ukL2_(Ω,Rn₎
for all u ∈ H1(Ω), (1.1)

where C = C(Ω) > 0 is a constant depending on the domain Ω. This result can be proved via a simple integration by parts and Cauchy-Schwarz inequality if the domain is the upper graph of a Lipschitz function, i.e.,

Ω =x = (xh, xn) ∈ Rn−1× R; xn> ω(xh) (1.2)

where ω : Rn−1_{→ R is a globally Lipschitz function.}

2

The result

Let Ω ⊂ Rn _{be a domain of the form (1.2). The exterior unit normal ν of Ω at a point x = (x}

h, ω(xh)) on the boundary Γ of Ω: Γ :=x = (xh, xn) ∈ Rn−1× R; xn= ω(xh) is given by ν(xh, ω(xh)) = 1 p1 + |∇hω(xh)|2 (∇hω(xh), −1)

(∇h denotes the “horizontal gradient” on Rn−1 acting on the “horizontal variable” xh). We denote by

θ∈ [0,π 2) the angle θ= arccos inf xh∈Rn−1 1 p1 + |∇hω(xh)|2  , (2.1)

so that in particular for e = (0_Rn−1,1) the “vertical” direction, we have

−e · ν(xh, ω(xh)) =

1

p1 + |∇hω(xh)|2

≥ cos θ > 0, for all xh∈ Rn−1. (2.2)

Theorem 2.1. Let _{Ω ⊂ R}n _{be as above. Letϕ}

: Rn

→ R be a smooth function with compact support. Then Z

Γ|ϕ|

2_{dσ ≤} 2

cos θkϕkL2(Ω)k∇ϕkL2(Ω,Rn), (2.3) whereθ has been defined in (2.1).

Proof. Let ϕ : Rn

→ R be a smooth function with compact support, and apply the divergence theorem in Ω with u = ϕ2_{ewhere e = (0} Rn−1,1). Since div (ϕ 2_{e) = 2 ϕ (e · ∇ϕ), we obtain} Z Ω2 ϕ (e · ∇ϕ) dx = Z Ω div (ϕ2e) dx = Z Γ ν_{· (ϕ}2e) dσ. Therefore using (2.2) and Cauchy-Schwarz inequality, we get

cos θ Z Γ ϕ2dσ ≤ −2 Z Ω ϕ(e · ∇ϕ) dx ≤ 2 kϕkL2(Ω)k∇ϕkL2_(Ω,Rn₎,

which gives the estimate (2.3).

Corollary 2.2. There exists a unique operator T _{∈ L (H}1(Ω), L2(Γ)) satisfying T ϕ= Tr|Γϕ, for allϕ∈ H

1_{(Ω) ∩ C (Ω)}

and

kT kL(H1_(Ω),L2_(Γ)) ≤√1

cos θ. (2.4)

Proof. The existence and uniqueness of the operator T follow from Theorem 2.1 the density of C∞ c (Ω) in

H1(Ω) (see, e.g., [1, Theorem 4.7, p. 248]). Moreover, (2.3) implies kϕk2L2(Γ,dσ)≤ 1 cos θ kϕk 2 L2(Ω)+ k∇ϕk2L2_(Ω,Rn₎
, for all ϕ ∈ C_c∞(Ω), which proves (2.4).

References

[1] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987, Oxford Science Publications.

[2] Jindˇrich Neˇcas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012, Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordi-nation and preface by ˇS´arka Neˇcasov´a and a contribution by Christian G. Simader.