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Accademia Nazionale delle Scienze detta dei XL
Memorie di Matematica e Applicazioni
123ë (2005), Vol. XXIX, fasc. 1, pagg. 149-152
MAURIZIO CHICCO (*)
______
Harnack Inequality for Solutions of Mixed Boundary
Value Problems Containing Boundary Terms
ABSTRACT. Ð We prove Harnack inequality near the boundary for solutions of mixed boundary
value problems for a class of divergence form elliptic equations with discontinuous and unbounded
coefficients, in presence of boundary integrals.
1. - INTRODUCTION
In this note we prove Harnack inequality, near the boundary of V, of the solutions of a
mixed problem for a class of divergence form elliptic equations, containing integral terms
on the boundary.
Let V be an open subset V of Rn , and V be the subspace of H 1 (V) defined by
(1)
V :ˆ fv 2 H1 (V) : v ˆ 0 on G o in the sense of H1 (V)g
where G o is a closed (possibly empty) subset of @V. Consider furthermore the bilinear
form
…
…n X
n
n
o
X
(2)
aij uxi vxj ‡
(bi uxi v ‡ di uvxi ) ‡ cuv dx ‡ guv ds
a(u; v) :ˆ
V
i; jˆ1
iˆ1
G
where G :ˆ @VnG o .
Let u 2 V be a solution of the equation
…
…n
n
o
X
(3)
fi vxi dx ‡ hv ds
fo v ‡
a(u; v) ˆ
V
iˆ1
8 v 2 V:
G
We can remark that, if the functions we consider are sufficiently regular (for example of
(*) Indirizzo dell'Autore: Dipartimento di Ingegneria della Produzione, Termoenergetica e
Modelli Matematici, UniversitaÁ di Genova, Piazzale Kennedy Pad. D, 16129 Genova, Italia.
Ð 150 Ð
class C 1 (V)), as well as G, then u is a solution of the following problem:
8
n
X
>
>
>
Lu
ˆ
f
( fi )xi in V;
o
>
>
>
>
iˆ1
<
n
n
n
X
X
X
a
u
N
‡
d
N
u
‡
gu
ˆ
h
‡
fi Ni on G;
>
ij
x
j
i
i
i
>
>
>
i; jˆ1
iˆ1
iˆ1
>
>
>
:
u ˆ 0 on G o
where N is the normal unit vector to G (oriented towards the exterior of V) and L the
operator defined by
n
n h
n
n
h
i
i
X
X
X
X
Lu :ˆ
aij uxi xj ‡
(aij )xj uxi ‡ c
(di )xi u
bi di
i; jˆ1
iˆ1
jˆ1
iˆ1
In a former work [1] we proved HoÈlder regularity up to the boundary for solutions of
equation (3), under suitable hypotheses on the coefficients of the bilinear form a(:; :) and
known terms fi (i ˆ 0; 1; 2; ; n) and h.
In this note we prove Harnack inequality near the part G of the boundary for solutions
of equation (3), following, as in [1], the classical work by Stampacchia [4]
2. - NOTATIONS AND HYPOTHESES
Let V be an open bounded subset of Rn (with n 3 for simplicity); for the definition
of the spaces H 1;p (V) we refer for example to [2], [3].
P
aij ti tj yjtj2 8t 2 Rn a.e. in V, with y
Let us suppose aij 2 L1 (V) (i; j ˆ 1; 2; ; n);
n
p
a positive constant; bi 2 L (V); di 2 L (V); (i ˆ 1; 2; ; n); c 2 Lp=2 (V); g 2 Lp (G) with
p > n; p :ˆ p(n 1)=n.
Let G o be a closed (possibly empty) subset of @V, and define G :ˆ (@V)nG o . We
suppose that G is representable locally as the graph of a Lipschitz function: for the precise
hypotheses see [1], par. 3.
3. - MAIN RESULTS
THEOREM 3.1: Suppose that all the hypotheses listed in the previous paragraph are verified; let u 2 V be a non negative solution of the equation
a(u; v) ˆ 0
8v 2 V
Then there exist two positive constants K1 and r, depending on the coefficients of a(:; :) and
on the regularity of G, such that for any x 2 G and any r with 0 < r r it turns out
ess sup u K1 ess inf u
V(x;r)
where V(x; r) :ˆ fx 2 V : jxi
V(x;r)
xi j < r; i ˆ 1; 2; ; ng.
Ð 151 Ð
PROOF: As in [1], it is convenient to follow the proof e.g. of Theorem
8.2 of [4],
„
remarking only the differences due to the presence of the new term guv ds in the
G
definition of the bilinear form a(:; :) (see (2)).
Let us begin with the proof of Lemma 8.1 of [4]. Proceding in the same manner, we
find„ in the right member of the last inequality (at the end of page 239) the new term
jpj jgja2 v2 ds. By using the inequality (33) of [1] we get at once
G
…
(5)
jgja2 v2 ds K42 kgkLn
1 (G\Q)
k(av)x k2L2 (V\Q)
G\Q
where we have denoted for brevity Q :ˆ fx 2 Rn : jxi xi j < r; i ˆ 1; 2; ; ng and
a 2 Co1 (Q). Note that in (5) the quantity kgkLn 1 (G\Q) can be made as small as we like by a
suitable choice of r. Therefore, combining (5) and the proof of Lemma 8.1 in [4], we can
bring the term kavx k2L2 (V\Q) to the left member of the last inequality of page 239 of [4], so
the conclusion follows as in [4].
As what concerns
Lemma 8.2 of [4], even in this case we find in the inequality (8.8)
„
the new term jgja2 ds. In order to evaluate this integral, we can use (5) where we put
G
v ˆ 1. Since it is a(x) ˆ 0 if x 2
= Q(x; 2r), jax j 2=r (see [4]), we easily deduce
…
jgja2 ds Ko kgkLn
1 (G\Q)
rn
2
G
where Ko depends only on K4 and n; the rest of the proof continues as in [4].
Also the remaining part of the proof of Theorem 8.1 of [4] can be followed in our new
situation. There are few differences, for example in the proof of Lemma 8.4 (page 241 at
the end) we must use Theorem 2 of [1] instead of the Remark 5.1 of [4]. The simple
details can be left to the reader.
p
COROLLARY 1. Suppose that all the hypotheses listed in the previous paragraph are
verified; let u 2 V be a non negative solution of the equation (3). Then there exist two
positive constants K2 and r, depending on the coefficients of a(:; :) and on the regularity of G,
such that for any x 2 G and any r with 0 < r r it turns out
ess sup u K2 ess inf u ‡
V(x;r)
V(x;r)
‡ K2
n
X
iˆ1
!
k f1 kLp (V) ‡ k fo kLnp=(n‡p) (V) ‡ khkLp (G) r 1
n=p
PROOF: We can proceed again as in [4] (Theorem 8.2), where we must use Lemma 3 of
[1] instead of Theorem 5.2 of [4].
p
Ð 152 Ð
COROLLARY 2. Suppose that all the hypotheses listed in the previous paragraph are
verified; let u 2 V be a non negative solution of the equation
a(u; v) ˆ 0
8v 2 V
Let G be an open subset of V such that G \ @V G. Then there exists a positive constant
K3 , depending on G; G and the coefficients of a(:; :), such that it turns out
ess sup u K3 ess inf u
G
G
PROOF: As in [4] it is possible to deduce our assertion from the previous Theorem, by
covering G by a finite number of sufficiently small cubes. In particular we must cover
G \ @V ˆ G \ G by small cubes with centres on G, in such a way as to apply the
Theorem to each of them, while we can apply Theorem 8.1 of [4] to the cubes contained
in V. The details of the proof can be left to the reader.
p
REFERENCES
[1] M. CHICCO - M. VENTURINO, HoÈlder regularity for solutions of mixed boundary value problems
containing boundary terms, Boll. Un. Mat. Ital. (to appear).
[2] E. GAGLIARDO, ProprietaÁ di alcune classi di funzioni in piuÁ variabili, Ricerche Mat., 7 (1958),
102-137.
[3] O. A. LADYZHENSKAYA, N. N. URAL8TSEVA, Linear and quasilinear elliptic equations, Academic
Press, New York (1968).
[4] G. STAMPACCHIA, Le probleÁme de Dirichlet pour les eÂquations elliptiques du second ordre aÁ
coefficients discontinus, Ann. Inst. Fourier, Grenoble 15 (1965), 189-258.
__________________________________________________________________________________________
Direttore responsabile: Prof. A. BALLIO - Autorizz. Trib. di Roma n. 7269 dell'8-12-1959
«Monograf» - Via Collamarini, 5 - Bologna