Proof of Harnack’s inequality for solutions to the mean curva- curva-ture equation and proof of a boundary Harnack’s inequality

Choose hl4hl(r)C¹(B_R(x₀) ), r4dist (x₀,x₁), with

. /

´

0Gh_lG1 ,

h_l41 in B_lR(x₀), h_l40 on ¯BR(x₀), (3.1)

for some l, 0ElE1 , such that 1

( 12l)R G NDh_l(r)N G 11d₀ ( 12l)R , (3.2)

for some d0, 0Ed0E1 and for lRGrGR. Thus 12 r

R Gh_l(r)G12 ( 11d0)r

R ,

(3.3)

for lGrGR.

Suppsose that uC²(V) is a solution to (0.1) in V. Taking h4 4h_l(u2mR) and h4h_l(MR2u) in (0.12), we obtain

B_R(x₀)

NDuN

k

11 NDuN²

QDhdx1n

B_R(x₀)

Hhdx40 ,

which yields

(3.4)

B_lR(x₀)

NDuN²

k

11 NDuN² dxG

G

B_R(x₀)0B_lR(x₀)

NDh_lN(u2mR)dx2n

BR(x0)

Hh_l(u2mR)dx4

4

BR(x0)0BlR(x0)

NDh_lN(u2m_R)dx2n

BR(x0)0BlR(x0)

Hh_l(u2m_R)dx2

2n

B_lR(x₁)

H(u2m_R)dx,

and

(3.5)

BlR(x₀)

NDuN²

k

11 NDuN² dxG

G

BR(x0)0BlR(x0)

NDh_lN(M_R2u)dx2n

B_lR(x₀)

Hh_l(M_R2u)dx4

4

BR(x0)0BlR(x0)

NDh_lN(M_R2u)dx2n

BR(x0)0BlR(x0)

Hh_l(M_R2u)dx2

2n

B_lR(x₀)

H(M_R2u)dx.

By (3.1) and (3.2), we have

BR(x0)0BlR(x0)

NDh_lN(M_R2m_R)dxGnv_n(M_R2m_R)

g

( 1¹2¹l)^d0R

h

lR R

rⁿ²¹dr4

4

g

¹1¹2^dl⁰

h

^{( 12ln)(MR2mR)vnRn21.}

Since d0 can be arbitrarily small, we have

(3.6)

B_R(x₀)0B_lR(x₀)

NDh_lN(M_R2mR)dxG

g

¹1²2^llⁿ

h

^{(MR2mR)vnRn21.}

3.1. Proof of theorem 3.

We also have

n

BR(x0)0BlR(x0)

Hh_l(M_R2m_R)dxF

Fn²( inf

xBR(x0)H)vn(M_R2m_R)

y

lR

R

g

^{12 ( 11}R^d0)r

h

^rn21dr

z

Fn²( inf

xBR(x0)H)(M_R2m_R)

g

¹²n^ln 212lⁿ¹¹

n11

h

^vnRn.

Since d0 can be arbitrarily small, we obtain

(3.7) n

B_R(x₀)0B_lR(x₀)

Hh_l(MR2mR)dxF

Fn²( inf

xBR(x0)H)(M_R2m_R)

g

¹²n^ln 212lⁿ¹¹

n11

h

^vnRn21.

Moreover, we have

n

B_lR(x₀)

H(M_R2m_R)dxFn( inf

xB_R(x₀)H)(M_R2m_R)lⁿv_nRⁿ. (3.8)

From (3.6), (3.7),(3.8) and (0.39), we obtain

BR(x0)0BlR(x0)

NDh_lN(M_R2m_R)dx2

2n

BR(x0)0BlR(x0)

Hh_l(M_R2m_R)dx2n

B_lR(x₀)

H(M_R2m_R)dxG

G(M_R2m_R)vnRⁿ²¹Q

Q

k g

¹1²2^llⁿ

h

^{2n2( infxBR(x0)H)R}

g

¹²n^ln 2 12lⁿ¹¹

n11 1lⁿ

hl

⁴

4(M_R2m_R)vnRⁿ²¹Q

Q

g

¹1²2^llⁿ

h k

^{12n( infxB}R(x₀)H)R( 12l)

g

¹²n1ⁿ1

12lⁿ¹¹

12lⁿ 1 lⁿ 12lⁿ

hl

⁴

4(MR2mR)vnRⁿ²¹C_lQ

Q

k

^{12( infxB}R(x₀)H)R

g

n1ⁿ 1

h

^{( 12l)}

g

^12nlC_lⁿ

h

^{2( infxBR(x0)H)R}

g

^nlC_lⁿ

h l

⁴

4(MR2mR)vnRⁿ²¹C_lQ

Q

k

^{12( infxBR(x0)H)R}

g

n1ⁿ1

h

^{( 12l)}

g

^11lCⁿ_l

h

^{2n(infxBR(x0)H)Rln}C¹_l¹

l

^.

Since there holds either M_R2m_RG2(u(x₁)2m_R) or M_R2m_RG G2(M_R2u(x₁) ), we have either

BR(x0)0BlR(x0)

NDh_lN(u2m_R)dx2

2n

BR(x0)

Hh_l(u2mR)dxG2(u(x₁)2mR)C_lC_l*vnRⁿ²¹,

or

BR(x0)0BlR(x0)

NDh_lN(M_R2u)dx2

2n

B_R(x₀)

Hh_l(M_R2u)dxG2(M_R2u(x₁) )C_lC_l*vnRⁿ²¹.

where C_l and C_l* are given respectively in (0.39) and (0.40). Inserting these into (3.4) or (3.5), and subsequently insert what results in into (0.5) or (0.6), we obtain (0.37) and (0.38).

Since there also holds either M_R2m_RG2(u(x₀)2m_R) or M_R2 2mRG2(MR2u(x0) ), we have, either

BR(x0)0BlR(x0)

NDh_lN(u2m_R)dx2

2n

B_R(x0)

Hh_l(u2mR)dxG2(u(x0)2mR)C_lC_l*vnRⁿ²¹,

or

BR(x0)0BlR(x0)

NDh_lN(M_R2u)dx2

2n

BR(x0)

Hh_l(M_R2u)dxG2(M_R2u(x₀) )C_lC_l*vnRⁿ²¹.

Inserting these into (3.4) or (3.5), and subsequently insert what results in into (0.5) or (0.6), we obtain (0.42) and (0.43) under the hypothesis of (0.41).

3.2. Proof of theorem 6.

Since there holds either MR2mRG2(u(x1)2mR) or MR2mRG G2(M_R2u(x1) ), we obtain from (3.6) that either

BR(x0)0BlR(x0)

NDh_lN(u(x₀)2m_R)dxG

G2

g

¹1²2^llⁿ

h

^{(u(x1)2mR)vnRn2142Cl(u(x1)2mR)vnRn21,}

or

B_R(x₀)0B_lR(x₀)

NDh_lN(MR2u(x0) )dxG

G2

g

¹1²2^llⁿ

h

^{(u(x1)2mR)vnRn2142Cl(MR2u(x1) )vnRn21,}

Inserting these into (3.4) or (3.5), and subsequently insert what results in into (0.5) or (0.6), we obtain (0.48) and (0.49).

Since there also holds either M_R2m_RG2(u(x₀)2m_R) or M_R2 2m_RG2(M_R2u(x₀) ), we have, either

BR(x0)0BlR(x0)

NDh_lN(u(x₀)2m_R)dxG

G2

g

¹1²2^llⁿ

h

^{(u(x0)2mR)vnRn2142Cl(u(x0)2mR)vnRn21,}

or

BR(x0)0BlR(x0)

NDh_lN(M_R2u(x₀) )dxG

G2

g

¹1²2^llⁿ

h

^{(MR2u(x0) )vnRn2142Cl(MR2u(x0) )vnRn21.}

Inserting these into (3.4) or (3.5), and subsequently insert what results in into (0.5) or (0.6), we obtain (0.51) and (0.52) under the hypothesis of (0.50).

Appendix. Proof of Proposition 3 and Proposition 4..

The equation (0.1) is the Euler equation of the functional F_*(v)4

V

k

11 NDvN²dx1n

V

0 v

H(x,t)dt dx.

And corresponding to the Dirichlet problem with boundary data c and the capillarity problem with boundary contact angle u are the problems of minimizing the respective functionals

F_*^(v)₁

¯V

Nv2cNdHn21

and

F_*^(v)1

¯V

( cosu)v dHn21

among all vBV (V), where Hk is the k-dimensional Hausdorff mea-sure.

Alternatively, we consider the problem of minimizing the functional F^(v)₄

V

k

11 NDuN²dx1

V

0 v

H(x,t)dx dt1

¯V

k(x,v)dHn21, with

k(x,v)4

0 v

g(x,t)dt. For the capillarity problem, we have

g(x,t)4cosu and

k(x,t)4

0 v

cosudt. For the Dirichlet problem, we have

g(x,t)4122f_C(x,t)

and

k(x,u)4 Nu2f(x)N 2 Nf(x)N.

Here and throughout this section,f_Vis the characteristic function of the subgraph V of v:

fV(x,t)4^./

´ 1 , 0 ,

if tEv(x), if tFv(x).

M. Miranda [22] introduced the notion of generalized solutions for the minimal surface equation and used it successively both in the Dirich-let problem in infinite domains [22] and in the problem of removable sin-gularities [23]. E. Giusti in [11] and [12] used the same notion of gen-erailized solutions respectively in the problem of maximal domains for the mean curvature equation and boundary value problems for the mean curvature equation.

The idea of generalized solutions originates from the observation that a function u:VOR is a solution of (0.1) in V if and only if its subgraph

U4 ](x,t)V3R,tEu(x)( minimizes the functional

F*(U)4

V3R

NDf_UN 1n

V3R

Hf_Udx dt

locally inV3R, in the sense that for every setVcoinciding withU out-side some compact set K%V3R, we have

K

NDfUN 1n

K

HfUdx dtG

K

NDfVN 1n

K

HfVdx dt.

Moreover, a functionuBV (V) minimizes FⁱⁿV if and only if its sub-graph minimizes the functional

F(U)4

V3R

NDfUN 1n

V3R

HfUdx dt1

¯V3R

gfUdHn.

Minimization is here to be understood in the following sense: forTD0 , set

QT4V3[2T,T] , dQT4¯V3[2T,T] ,

and for U%Q, FT(U)4

QT

NDf_UN 1n

QT

Hf_Udx dt2

dQT

gf_UdHn.

We say that U minimizes F_T in Q_T if F_T(U)GF_T(S)

for every Caccioppoli setS%Q_T. We say thatU minimizesFinV3R if U minimizes FT in QT for every TD0 .

DEFINITION (Miranda[22]).

(1) A function u:VO[2Q,Q] is a generalized solution of the equation(0.1)inVif its subgraph U minimizes the functionalF* local-ly in V3R.

(2) A function u: VO[2Q,Q] is a generalized solution for the function F if its subgraph U minimizes F in Q.

We note that a generalized solution can take the values6Qon a set of positive n-dimensional Hausdorff measure. However, it follows from Miranda [21] that if a generalized solutionu(x) can be modified on a set of zero n-dimensional Hausdo rff measure to be locally bounded, then u(x) is a classical solution of (0.1) in V.

Proposition 5 below is derived in the proof of Theorem 1.1 of Giusti [12]. The special case in Proposition 6 below wherem41 and¯VC²are shown by the proof of Theorem 3.2 of [12]. Proposition 6 is fully estab-lished in Lemma 7.6 of Finn [3].

It is easy to see that Proposition 3 and Proposition 4 follow immedi-ately from Proposition 5 and Proposition 6, together with a subsequent consideration of U8 instead of U.

PROPOSITION 5. Let U minimize F

* locally in Q4V3R. If z04 4(x₀,t₀) is a point in Q and if for all rD0 we have

NU_r(z₀)ND0 ,

then, there exist positive constants C₀and R₀,depending only on n and inf

Q H such that

NUr(z0)N FC0rⁿ¹¹, (A.1)

for every rGmin (R₀, dist (z₀,¯Q) ), where we set U_r(z0)4UOC_r(z0) , with

C_r(z₀)4 ]z4(x,t) :Nx2x₀N Er,Nt2t₀N Er(. In particular, we can take

C04 1

4(n11 )k_{(n11 )}, (A.2)

and

R04

. /

´

u

^2nk(n)vnN¹inf

Q H(x,t)N

v

^{1 /n}

Q,

if inf

Q H(x,t)E0 , if inf

Q H(x,t)F0 , (A.3)

where we denote k_(m) the isoperimetric constant in R^m, mF1.

PROPOSITION 6. Suppose that there exist constants Q₀D0 and g×, 0Gg×E1 , such that

g(x,t)F 2g×, for all x¯V and tDu0.

Suppose further that for some constantm,with m g×E1 and C_V depend-ing only on V, an inequality

¯V

NvNdxGm

V

NDvNdx1C_V

V

NvNdx, (A.4)

holds for all vBV (V).Let U minimizes Fin Q4V3R,and let z04 4(x0,t0), t0Du011 , be a point of Q such that for every positive r

NU_rND0 ,

where U_r is defined as in Proposition 1. Then there exist constants R₁D0 and C₁D0 determined completely by n, inf

Q H(x,t), g×, m and C_V such that

NUrN FC1rⁿ¹¹, for every rGR1. (A.5)

In particular, we can take

C14 12g× 16(n11 )k_{(n11 )} . (A.6)

and if inf

Q H(x,t)F0 ,

R₁4

u

_CVk(n ^Cg×*

11 )( 2vn)^{1 /(n11 )}

v

^,

(A.7)

where we set

C_g×*4min

g

¹2 , 12( 2m21 )g× 3g×11

h

^;

if inf

Q H(x,t)E0 , we firstly take R×

1 so small that R×

1G

u

^2(m₁^{11 )2nk( 2m}(n)²v^{1 )}nN^g×inf

Q HN

N v

^{1 /n,}

and then take

R₁4min

u

_{CVk(n11 )}^C_{( 2}^g×**_vn)^{1 /(n11 ) ,R×1}

v

^,

(A.8)

where we set

C_g×**4min

u

¹₂ ^{, 12( 2m21 )g×2(m1}_3g×^{1 )nk(n)NinfQ HNvn(R×1)n}

11

v

^.

We notice that Lemma 1.1 in Giusti [11] established (A.4) form41 in the special case that¯VC²and we have formulated this result as Lem-ma 1 in 0.2 of our present work.

An inequality of the form (A.4) appears first in Emmer [2], with m4

k

11L²

for any Lipschitz domain with Lipschitz constantL. (See also [19, page 203]). On pages 141-143 of Finn [3], this result is extended to include do-mains in which one or more corners with inward opening angle appear.

As pointed out on page 197 of [3], this extended result permits inward cusps and even boundary segments that may physically coincide but are

adjacent to different parts ofV. However, it is pointed out on page 143 of [3] that an outward cusp or a vertex of an outwa rd corner is not permitted.

We end this section with a sketch of the reasoning in [3] and [12]

which leads to Propositions 5 and 6, mainly for the purpose of unifying the notation designations in [3] and [12]. Indeed, from comparing the values of the functionalsF*andFtaken byUwith those taken byU0C_r,

for almost all r, the previous two inequalities lead respectively to (A.9)

_NDfU

the curvature term can be estimated as follows:

again by the isoperimetric inequality , F 1

This leads to the estimate (A.1) withC₀ taken as in (A.2), wheneverrE Emin (R₀, dist (z₀,¯Q) ), with R₀ given in (A.3).

In caserFdist (z₀,¯Q), we have to handle the third term on the right

hand side of (A.10). By (A.4) and the isoperimetric inequality, the last inequality leads to

(A.13)

This and the last identity yield

_NDfU

From (A.11), (A.13) and (A.15), we obtain (A.16) n

Choosing rso small that (A.7) is satisfied if inf

Q HF0 and (A.8) is satis-fied if inf

Q HE0 , we know that (A.14) is satisfied and the right hand side

of (A.16) is bounded below by

g

^{2( 11}2^g×)

h

_Q

^s

^NDfUrN. Inserting this into (A.10), we obtain

¯C_r

f_UdHnF12g×

4

Q

NDf_UrN. (A.17)

From this, (A.14), (A.15) and the isoperimetric inequality, we obtain d

drNUrN 4

¯Cr

fUdHnF 12g×

16

_NDfU

rN F 12g× 16

1

k_{(n11 )}NU_rN

n n11. This leads to the estimate (A.5) withC1taken as in (A.6) and completes the proof of Proposition 6.

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Manoscritto pervenuto in redazione il 25 novembre 2002.

Dans le document Harnack's inequalities for solutions to the mean curvature equation and to the capillarity problem (Page 26-40)