Some existence results on the exterior Dirichlet problem for the minimal hypersurface equation

www.elsevier.com/locate/anihpc

Some existence results on the exterior Dirichlet problem for the minimal hypersurface equation

Nedir do Espírito-Santo

, Jaime Ripoll

aUFRJ, Instituto de Matemática, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ, Brazil bUFRGS, Instituto de Matemática, Av. Bento Gonçalves 9500, 91540-000 Porto Alegre, RS, Brazil Received 22 September 2010; received in revised form 12 January 2011; accepted 6 February 2011

Available online 16 March 2011

Abstract

It is proved the existence of solutions to the exterior Dirichlet problem for the minimal hypersurface equation in complete non- compact Riemannian manifolds either with negative sectional curvature and simply connected or with nonnegative Ricci curvature under a growth condition on the sectional curvature.

©2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

With the development of the Riemannian geometry many PDE results proved in the Euclidean geometry for the class of geometric operators have been investigated in more general Riemannian manifolds. In the case of the Lapla- cian equation, S.T. Yau proved that Liouville’s theorem (an entire nonconstant solution of the Laplace equation inRⁿ is necessarily unbounded) is also true in a complete Riemannian manifoldMⁿwith nonnegative Ricci curvature and conjectured that ifMis simply connected with sectional curvature satisfying−κ₁²KM−κ₂²<0,for some con- stantsκ1,κ2,then there should exist bounded solutions of the Laplacian equation onM. This has been proved in the 80’s by Anderson and Sullivan (see [12]). Actually, they proved that on the compactified manifoldM:=M∪∂_∞M one can solve the Dirichlet problem for the Laplace equation for any given continuous data on∂_∞M.

For the case of the minimal surface equation, it is known that Bernstein’s theorem is true in a complete noncompact 2-dimensional Riemannian manifold with nonnegative curvature: Any entire minimal graph inM²is totally geodesic inM²×R[10]. As far as the authors know, if a similar result holds in higher dimensions is still an open question.

In the case of simply connected Riemannian manifolds with negative curvature one should expect a similar existence result for minimal graphs as Anderson and Sullivan results for harmonic functions. In fact, in [4] the authors prove the existence of entire solutions of the minimal hypersurface equation with any given smooth data at the asymptotic boundary∂_∞Mof M ifMis complete, simply connected, with sectional curvature satisfying K_M−k²<0 and such that isotropy subgroup of the isometry group of M at some point p of M acts transitively on the geodesic spheres centered atp(this result has a similar harmonic counterpart due to H.I. Choi, see [2, Theorem 3.6]). In [6]

* Corresponding author.

E-mail addresses:[email protected] (N. do Espírito-Santo), [email protected] (J. Ripoll).

0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2011.02.007

A. Gálvez and H. Rosenberg prove the existence of solutions when dimM=2 assuming only a negative upper bound for the sectional curvature ofM. A partial extension of this last result and an improvement of the one of [4] (and of Theorem 3.6 of Choi [2] too) was obtained in [11], where an unified treatment for the harmonic and minimal case is given. In [8] it is investigated the extension to a 2-dimensional Riemannian manifold of the classical result of Jenkins and Serrin on the Dirichlet problem for the minimal surface equation on bounded domains with infinity boundary value data.

Another result on minimal graphs, due to N. Kutev and F. Tomi, asserts the existence of solutions to theexterior Dirichlet problem for the minimal surface equation, namely: IfΩ⊂R²is a domain such thatR²\Ωis bounded, then there is a nonzero solution of the minimal surface equation inΩassuming at∂Ωany given boundary data with small enough oscillation (see Theorem E of [7] for the precise statement). In the present article we investigate the extension of Kutev and Tomi’s result to a Riemannian manifold in the case of vanishing boundary data. Precisely, we study the existence of solutions to the following Dirichlet problem:

⎧⎨

⎩M(u):=div

gradu 1+ |gradu|²

=0 inΩ, u∈C²(Ω)∩C⁰(Ω), u|∂Ω=0

(1) whereΩ is an unbounded domain (open and connected) in a complete noncompact Riemannian manifoldMsuch that∂Ωis compact; div and grad are the divergence and gradient inM.

We prove existence results of solutions of (1) in the cases that eitherMhas nonnegative Ricci curvature orMis simply connected and with sectional curvatureK_M satisfyingK_M−k²<0 for some positive constantk.In the case of negative curvature, we prove:

Theorem 1.Assume thatMis simply connected and that the sectional curvatureK_M ofMsatisfiesK_M −k²<0 for some positive constantk.We require thatΩ is a domain ofM satisfying the exterior sphere condition, namely, given p∈∂Ω,there is a geodesic sphere ofMpassing throughp,tangent to∂Ω atp which is the boundary of a geodesic ball containingΩ.

Given any nonnegative real numbers,there exists a bounded solutionu∈C^∞(Ω)∩C⁰(Ω)of(1)such that

x→lim∂Ωsupgradu(x)=s and

maxΩ |u|2s k .

We next consider the case thatMhas nonnegative Ricci curvature. We prove:

Theorem 2.LetMbe ann-dimensional, complete noncompact Riemannian manifold with nonnegative Ricci curva- ture and with sectional curvature satisfying the growth condition

K_M(x) 4c²

(1+4c²ρ²(x))², x∈M,

for some constantc >0,whereρis the distance function to a totally geodesic submanifoldSofM, ρ(x)=inf d(x, y)y∈S

,

d=Riemannian distance in M, and K_M(x)is the maximum of the sectional curvature of M on planes of T_xM containinggradρ.LetΩ be an unbounded domain inMsuch that∂Ωis compact and assume thatρis smooth onΩ. Then, for any given nonnegative real numbers,there exists a solutionu∈C^∞(Ω)∩C⁰(Ω)of(1)such that

x→lim∂Ωsupgradu(x)=sup

Ω |gradu| =s. (2)

We observe that the existence ofSis part of the hypothesis of Theorem 2. However, ifMhas nonnegative sectional curvature the existence ofSis guaranteed by the Soul Theorem of Cheeger and Gromoll [1]. We remark that Cheeger and Gromoll’s theorem does not apply if it is only assumed that the Ricci curvature ofMis nonnegative.

We finally remark that the vanishing boundary data hypothesis of the exterior Dirichlet problem, at least in The- orem 2, cannot be dispensed, since Theorem N of [7] proves the existence, whenM=R²,of continuous nonzero boundary data with arbitrarily smallC⁰norm for which the exterior Dirichlet problem does not have a solution.

2. Some notations and preliminary results

It is assumed thatMis a complete Riemannian manifold. We shall make use later of the following computa- tion. LetNbe any given smooth compact submanifold ofMand denote byζ the distance toN,that is,

ζ (x)=d(x, N )=min d(x, y)y∈N

whered is the Riemannian distance inM. Given a functionϕ∈C²(R),setw=ϕ(ζ ).If ζ is differentiable in a neighborhood ofx∈M,considering an orthonormal basis inT_xMcontaining gradζ (x),we obtain

M(w)=div

gradw 1+ |gradw|²

=div

ϕ gradζ 1+(ϕ)²

, and, after some computations

M(w)(x)=(n−1)ϕ (ζ (x))+ϕ(ζ (x))(1+(ϕ(ζ (x)))²)ζ (x)

(n−1)(1+(ϕ(ζ (x)))²)^3/2 . (3)

The following result follows from Lemma 6 of [3]:

Lemma 3.LetΛbe aC^2,αbounded open subset ofMandu∈C^2,α(Λ)a solution ofM(u)=0inΛ. Assume thatu is bounded inΛand that|gradu|is bounded onΓ =∂Λ. Then|gradu|is bounded inΛby a constant that depends only onsup_Λ|u|andsup_Γ|gradu|.

In the case thatMhas nonnegative Ricci curvature we have in fact a maximum principle for the gradient. This principle is fundamental for the proof of Theorem 2:

Lemma 4.Assume that the Ricci curvature ofMis nonnegative. LetΛbe aC^∞bounded open subset ofM.Then any solutionu∈C^∞(Λ)ofM(u)=0inΛsatisfies the gradient maximum principle

maxΛ |gradu| =max

∂Λ |gradu|.

Proof. Letηbe a unit normal vector field orthogonal to the graphGofusuch thatη, ∂t0, ∂tbeing the unit vector in theRdirection. Since RicM×R0 it follows from Proposition 1 of [5] that

η, ∂t = −

RicM×R(η)+ B²

η, ∂t0,

whereBis the norm of the second fundamental form ofG. The functionη, ∂tis then superharmonic onGso that maxΛ |gradu| =min

G η, ∂t =min

∂Gη, ∂t =max

∂Λ |gradu|. 2

Notation.Under the hypothesis and notations of Theorem 1, we denote byγ the Riemannian distance inM to∂Ω restrict toΩ,namely

γ (p)=inf d(p, q)q∈∂Ω

, p∈Ω.

Moreover, we setFr =γ⁻¹(r), r0 and denote byHF_r the normalized mean curvature ofFr with respect to the unit vector field normal toFr pointing to the bounded connected component ofM\Fr.Note that

γ (x)=(n−1)HF_{γ (x)}(x), (4)

x∈M.

In the next lemma we use the exterior sphere condition to estimate from below the mean curvature of the level hypersurfacesF_r which is fundamental to the construction of low barriers for problem (1) in the case of Theorem 1:

Lemma 5.IfMis simply connected and with sectional curvature satisfyingK_M−k²<0, k >0, then infFr

H_Fr k (5)

for anyr >0.

Proof. Givenp∈F_r there isq∈∂Ω and a minimizing geodesicα:[0, r] →M such thatα(0)=q andα(r)=p.

LetSl(p0)be a geodesic sphere of Mwith some radiusl and some centerp0 passing throughq,which is tangent to∂Ω atq and such that∂Ω⊂B_l(p₀).Then, by Gauss Lemma,S_l+r(p₀)passes top, is tangent toF_r atp and F_r ⊂B_l+r(p₀).It follows from the tangency principle for constant mean curvature hypersurfaces and the Hessian Comparison Theorem thatH_Fr H_Sl+r(p0)k. 2

3. Proof of Theorem 1

We first assume thatΩis aC^2,αdomain, 0< α <1.Givens0, we look for constantsa, bandcsuch thatϕ(r)= (ar+b)/(r+c)determines a subsolution v_s(x):=ϕ(γ (x))satisfyingv_s|∂Ω=0 and |gradv_s|∂Ω=s.Obviously b=0 and|gradv_s|∂Ω=ϕ(0)=simplies thata=sc.Thenϕ(r)0 and it follows from (3), (4) and (5) thatv_s is a subsolution if

ϕ (r)+kϕ(r) 1+

ϕ(r)2

= sc² (c+r)⁶

kr⁴+2(2ck−1)r³+6c(ck−1)r²+2c²(2ck−3)r+c³

kcs²+kc−2

0

which is valid if we choosec=2/ksince then all the coefficients of the quartic polynomial onrbecome positive. We then proved that

vs(x)=ϕ γ (x)

= 2sγ (x) kγ (x)+2

is a subsolution satisfyingv_s|∂Ω=0 and|gradvs|∂Ω=s.Moreover

γ (x)lim→∞v_s(x)= lim

r→∞ϕ(r)=2s k .

Letm∈Nbe given,m1. SettingΩ_m=Ω∩ {x∈Ω|γ (x) < m},we prove the existence of a solutionw_m∈ C^2,α(Ω_m)ofM=0 inΩ_mwithw_m|∂Ω=0 and such that sup_∂Ω|gradw_m| =s.To this end, set

Tm=

t0∃ut∈C^2,α(Ωm)such thatM(ut)=0, ut|∂Ω=0, ut|Γ_m=t, sup

∂Ω

|gradut|s whereΓ_m=∂Ω_m\∂Ω.

We haveT_m= ∅,since 0∈T_m. We prove in the sequel thatT_mis bounded, that the supremum ofT_mis assumed and, what is the most fundamental point, that ift_m=supT_mthen sup_∂Ω|gradutm| =s. We begin by proving that if t∈T_mthen

tv_s|Γm= 2sm

km+2. (6)

Givenε >0,we first prove thatt < vs+ε|Γ_m.By contradiction, assume thattvs+ε|Γ_m.Since

|gradvs+ε|∂Ω=inf

∂Ω|gradv_s+ε| =s+ε >sup

∂Ω|gradu_t|

there is a neighborhood U of∂Ω inΩ such thatu_t(x) < v_s+ε(x)for allx∈U\∂Ω.It follows that there exists a domainU⊂V ⊂Ω_msuch thatu_t|∂V =v_s+ε|∂V,which is an absurd sincev_s+ε|V is a subsolution ofMand hence v_s+ε|V u_t|V.Lettingε→0 we havetv_s|Γm.It follows thatT_mis bounded and we may sett_m=supT_m<∞. We will prove thatt_m∈T_m.

We first prove ift∈Tm,then sup

∂Ω_m

|gradu_t|s. (7)

Assume thatt∈T_m.By definition ofT_mwe have sup_∂Ω|gradut|s.

Settingz_m=v_s|Γm we have, as proved above, thatz_m> t. Moreover, the functionw_s:=v_s−(z_m−t )is a subso- lution sincev_s is one,z_m−tis constant andM(w)does not depend onw,but just on its first and second derivatives.

Also,

w_s|∂Ω= −(z_m−t )0=u_t|Ω, w_s|Γm=t=u_t|Γm.

It follows that w_su_tt, w_s|Γm=t=u_t|Γm

and we may conclude that sup

Γm

|gradu_t|sup

Γm

|gradw_s| =ϕ(m)= 4s (km+2)² s proving (7).

Consider now a sequence{τ_j} ⊂T_mconverging tot_m asj goes to infinity. For eachj,there is a functionu_j ∈ C^2,α(Ω_m)such thatM(u_j)=0, uj|∂Ω=0 andu_j|Γm=τ_j.Since, by the maximum principle,

sup

Ωm

|u_j|t_m (8)

it follows from (7) and Lemma 3 that the sequence{u_j}has theC¹norm uniformly bounded inΩ_m.SinceΩ_m is a compactC^2,α domain, from elliptic PDE theory we haveC²compactness of{u_j}inΩ_m(see [9]). Hence, there is a subsequence of{u_j}that converges uniformlyC²inΩ_mto a solutionw_m∈C²(Ω_m)ofM=0 inΩ_m. From PDE elliptic regularityw_m∈C^2,α(Ω_m).

The functionw_mis a solution ofM=0 inΩ_mthat satisfiesw_m|∂Ω=0,w_m|Γm=t_mand sup_∂Ωm|gradwm|s.It follows thatt_m∈T_m,that is,w_m=u_tm.

We now note that sup_∂Ω|gradu_tm| =s. In fact: By contradiction, assume that sup_∂Ω|gradu_tm|< s.

Consider a functionφ∈C^2,α(Ωm)such thatφ|∂Ω=0 andφ|Γ_m=tm, set C₀^2,a(Ω_m)= ω∈C^2,α(Ω_m)ω|∂Ω_m=0

, and defineT:[0,2] ×C₀^2,α(Ω_m)→C^α(Ω_m)by

T (l, ω)=M(ω+lφ).

Then

T (1, ωm)=0

whereω_m=u_tm−φ.From elliptic PDE theory we have that the Fréchet derivative∂₂T (1, ωm)=dMwm is an iso- morphism (this is clear sincedMwm(h), h∈C₀^2,α(Ω_m),depends only on the first and second derivatives ofhand not onhand hence satisfies the maximum principle)so that, from the implicit function theorem, there exists a continuous function (on theC^2,α topology)i:(1−ε,1+ε)→C₀^2,α(Ωm),withi(1)=ωm such thatT (l, i(l))=0, l∈(1−ε, 1+ε).Therefore, since|gradut_m|∂Ω< sandwm=i(1)+φ,there existsl∈(1,1+ε)such that sup_∂Ω|i(l)+lφ|< s.

Since 0=T

l, i(l)

=M

i(l)+lφ ,

i(l)+lφ=0 at∂Ω andi(l)+lφ=lt_matΓ_m,we have thatlt_m∈T_m,contradiction sincelt_m> t_m=supT_m.Hence, sup_∂Ω|gradu_tm| =s.

Since, from (6), (7) and (8) max∂Ω_m|gradu_tm|ϕ(m)s, maxΩm

|u_tm| =t_mv_s|Γmϕ(m)2s k

are estimates that do not depend onmit follows from Lemma 4 that the sequence{u_tm}has uniformC¹estimates on compacts ofΩwhich implies the existence of a subsequence of{u_tm}converging uniformlyC^2,αon compacts ofΩ to a solutionu_s∈C^2,α(Ω)ofM=0 inΩsatisfyingu_s|∂Ω=0 and

sup

∂Ω

|gradu_s| =s.

Moreover, it also follows from the above estimates that there is a constantC(s)depending only onssuch that|us|1 C(s).IfΩis only aC⁰domain one considers a sequence ofC^2,αdomainsΛmsatisfying the exterior sphere condition such thatΩ⊂Λm+1⊂ΛmandΩ=

Λm.From what we have proved above, for eachmthere is a solutionvm∈ C^2,α(Λ_m)ofM(v_m)=0 such thatv_m|∂Λm=0 and

sup

∂Ω|gradv_m| =s|v_m|1C(s). (9)

From the Arzela–Ascoli theorem we may assume that v_m converges uniformly on compacts of Ω to a function u_s∈C⁰(Ω)such thatu_s|_∂Ω =0. IfU⊂Ω is compact,U open, then we have uniformC¹estimates ofv_m|U and then, from elliptic PDE theory,v_m|U contains a subsequence convergingC^∞tou_s onUso thatu_s|U∈C^∞(U )and M(u)=0 in U.By the diagonal method, we obtain the existence of subsequence of v_m converging to a solution u_s∈C^∞(Ω)∩C⁰(Ω)of (1). It is immediate to prove from (9) thatu_s satisfies (2). This concludes the proof of the theorem.

4. Proof of Theorem 2

We first consider the case thatΩ is aC^∞domain. Lets0 be given. Form >0 setBm= {ρm}and letm0be such that∂Ω⊂Bmfor allmm0.Givenm > m0,setΩm=Ω∩BmandSm=∂Ωm\∂Ω.

We will prove the existence ofm1> m0such that, givenmm1,there is a solutionum∈C^∞(Ωm)ofM=0 in Ω_mwithu_m|∂Ω=0 and sup_∂Ω|gradu_m| =s.To this end, givenm > m0,set

T_m=

t0∃u_t∈C^∞(Ω_m)such thatM(u_t)=0, sup

Ωm

|gradut|s, u_t|∂Ω=0, ut|Sm=t

.

It is clear thatT_mis bounded and the uniform bound for theC¹norm of the solutions ofM=0 corresponding to points ofT_mimply that the supremumt_m=supT_mis attained. Moreover, we may make use of the implicit function theorem, as in the previous theorem, to assert that the solutionu_m∈C^∞(Ω_m)ofM=0 inΩ_msuch thatu_m|Sm=t_m satisfies sup_Ωm|gradutm| =s.

The main part of the proof consists in proving that the maximum of|gradutm|is assumed at∂Ω.To this end, we construct barriers from above and from below tou_mwhich gradient less than or equal tos/2 atS_m.

Sincew_m:=t_mis a solution ofM=0 inΩ_mandu|∂Ω_mw_m|∂Ω_mwe haveu_mw_monΩ_mand hencew_mis an upper barrier.

To construct a barrier from below for (1) we first consider the paraboloid P of Rⁿ⁺¹ obtained by acting the rotational group of Rⁿ⁺¹, that leaves fixed the xn+1-axis, on the parabola (t,0, . . . ,0, ct²), t∈R. The sectional curvature of P aty=(y1, y2, . . . , yn+1)∈P with respect to any plane through the origin ofTyP that contains the tangent vector of the geodesic passing throughyand the vertex ofP is given by

KP(y)= 4c² (1+4c²r²(y))² wherer(y)=

y₁²+ · · · +y_n².

Letx∈Mandy∈P be such thatρ(x)=dP(y,0),wheredP is the Riemannian distance inP and 0 is the origin ofRⁿ⁺¹.Then

ρ(x)= r(y) 0

1+4c²t²dtr(y),

and, from the hypothesis, it follows that K_M(x) 4c²

(1+4c²ρ²(x))² 4c²

(1+4c²r²(y))² =K_P(y).

We then obtain, from the Hessian Comparison Theorem [12, Theorem 1.1], thatρ(x)ρP(y),whereρP(y)= dP(y,0).Therefore, ifϕis such thatϕ 0 it follows from (3) thatv(x)=ϕ(ρ(x))is a subsolution ofQinΩif

(n−1)ϕ ρ_P(y)

+ϕ ρ_P(y)

1+ ϕ

ρ_P(y)2

ρ_P(y)0, (10)

for ally∈P. We have

ρP(y)= n−1 r(y)

1+4c²r²(y). (11)

Introducing the notation δ(r)=

r 0

1+4c²t²dt andψ (r)=ϕ(δ(r))we have

ϕ δ(r)

=ψ(r) δ(r), ϕ

δ(r)

= 1 (δ(r))³

δ(r)ψ (r)−ψ(r)δ (r) .

Settingr=r(y)we have that the inequality (10) holds if and only if 1

(δ(r))³ (n−1)

ψ (r)δ(r)−ψ(r)δ (r)

+ψ(r) δ(r)2

+

ψ(r)2 ρ_P

0. (12)

Sinceδ(r)=√

1+4c²r²,δ (r)=4c²r/√

1+4c²r²,from (11) we have that (12) holds if, and only if, (n−1)

ψ (r)

1+4c²r²− 4c²rψ(r)

√1+4c²r²

+ψ(r)

1+4c²r²+

ψ(r)2

(n−1) 1 r√

1+4c²r² 0, that is

(n−1) r√

1+4c²r² r

1+4c²r²

ψ (r)−4c²r²ψ(r)+ψ(r)

1+4c²r² +

ψ(r)3

0

or r

1+4c²r²

ψ (r)+ψ(r)+ ψ(r)3

0.

We have that

ψ_a(r)=ψ (r)=√ a

r

√a 2c√

4−a

√4c²t²+1 4c²t²(4−a)−a

dt

is a solution of r

1+4c²r²

ψ (r)+ψ(r)+ ψ(r)3

=0 for alla∈ [0,4)and

r

√a 2c√

4−a. Moreover,ψ_asatisfies

ψ_a √

a 2c√

4−a

=0, ψ_a

√ a 2c√

4−a

= +∞. (13)

Letr0be such thatδ(r0)=m0anda0such that

√a₀ 2c√

4−a₀=r0, that is,

a0= 16c²r₀² 1+4c²r₀².

Setϕ(δ(r))=ψa₀(r).Givenm > m0,letrmbe such thatδ(rm)=m.Since ϕ(m)=ψ_a

0(r_m) δ(r_m) =

√a0

4c²r_m²(4−a₀)−a₀ we haveϕ(m)s/2 if and only if

√a₀

4c²r_m²(4−a₀)−a₀s

2 ⇔ rm

a₀(1+_s⁴2) 2c√

4−a₀ . Hence, if we choosem1> m0such that

r_m1

a₀(1+_s⁴2) 2c√

4−a₀ ,

sincer_mincreases withmwe haveϕ(m)s/2 for allmm₁.It follows from (13) that, for allmm₁, v_m(x):=

ϕ(ρ(x))is a “catenoid-like” subsolution ofMonΩ∩(B_m\B_m0).It satisfies atx∈Msuch thatρ(x)=m₀: v_m(x)=0,

gradv_m(x)= ∞ and, ifρ(x)=m,

gradv_m(x)s

2. (14)

Clearlyv_m+bis a subsolution ofMonB_m\B_m0for any constantband there isb₀such thatv_m(x)+b₀u_m(x) for allx∈Ω∩(B_m\B_m0).Set

b_m=max bv_m(x)+bu_m(x), ∀x∈B_m\B_m0 .

Since|gradvm(x)| = ∞forx∈S_m0=∂B_m0it follows from the maximum principle thatv_m(x)+b_m=t_mforx∈S_m andu_mv_monB_m\B_m0.Then

|gradum|Smmax |gradv_m|Sm,|gradwm|Sm

=max s

2,0

=s 2. It follows by Lemma 4 that

sup

∂Ω

|gradu_tm| =s.

Now, lettingm→ ∞,the uniformC¹estimates ofut_m on compacts ofΩ implies the existence of a subsequence of{u_tm}converging uniformly on compacts ofΩ to a solutionu_s∈C^∞(Ω)ofM=0 inΩ satisfyingu_s|∂Ω=0 and sup_∂Ω|gradus| =s.Note that for eachmthere is a pointq_m∈∂Ωsuch that|gradu_tm(q_m)| =s.Since∂Ω is compact a subsequence ofq_m converges toq∈∂Ω.It follows that|gradu_s(q)| =s,so thatu_s cannot be identically zero. If Ω is only aC⁰domain one considers a sequence ofC^∞domains converging toΩ similarly to what was done in the previous result. This concludes the proof of the theorem.

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