BULL. MATH, de la Soc. Sci. Math, de Roumanie Tome 36 (84), nr. 3-4 1992
LOCAL INVERSION THEOREMS AND IMPLICITE FUNCTIONS THEOREMS WITHOUT ASSUMING DIFFERENTIABILITY by M IH A I CRISTEA (RO M AN IA)
In the present paper, we shall generalize first a result of O.Martio, S.Rickmann and J.Vaisala from [10] page 12 (see also another proof in [7]). They proved that a function f : D C R n-*Rn which is differentiable on D such that det/ (x)^0 for every x ED, is a local homeomorphism on D, generalizing in this way the well-known local inversion theorem. We shall show that the local inversion theorem holds not only without assuming continuous differentiability, but even in some weaker conditions. We prove for instance that i f / : D C R n-*Rn is continuous such that for everyx&D there exists AxE L(Rnf t n) with det A x* 0 and c5>0 such that (f (z)-f (x)/lr(z—jc) ) > 0 for every zGD with z^x and ||z—;t||Rn. We define B^{xE:D \ f is not a local homeomorphism at*}. Definition. Let / \E-*F. We say that / is a light map if for every xE E and UE:V(x)ythere exists QEK(jc) such that Q C U and/ (x)£ f [F r Q ). 00
Definition. Let E be a metric space and 0 < p < 0 0 . Let m *(£) = inf^T d(Ai' f , i=1 00
where E = U A i and d(Ai)0
p-dimensional Hausdorff measure o f£ . Theorem A. [1], [13] page 13. Let D C R n be open, f:D - *R n continuous, open and discrete. Then d im B f< n —Z Theorem B. [6] Let D C R n be open, f : D -+Rn continuous and light and KCD closed and such that n\n_ 2(/(jC ))= 0 and f is a local homeomorphism an D\K.
Then f is a local homeomorphism on D. Theorem C. [3] Let E,F be n-manifolds of dimension n > 2 ,/: E -*F continuous, light such that there exists KC.D with d im f ( K ) < n —2 and such thatf is open in every point from D \K. Then fis open on D. Theorem D. [4], [5] Let DC.Rn be open, f : D-*Rn continuous and light such that there exists K d D with mn( f (K ))=0 and such that f is differentiable on D \K and
detf (x)>0 for every xGD \K or detf (x) 3 tD C R n be open, f :D->Rn continuous and light and 00
KCD such that K~ U p=1
with Kp closed sets and mn_ 2( f
for every pG N .
Local inversion theorem s and im plicite functions theorems
229
Suppose that / i s differentiable on D \K and det / (x)^0 for every xCD \K. Then / is a local homeomorphism onD . Theorem F. [6] Let D C R n be open, f :D->Rn be differentiable and 0 0 be such that \(f (x1y) —f (a, b),A z{y-b)) |< L for every
b). We can also suppose that ( f ( a ,y )- f (a,b),A z(y—b ) ) > 0 for y ^ b and \\y-b\\ rn for every
y&Rrn such that ||y —b ||=c3 and let M >0 be such that |\Az(y—b) ||< M for every „ 1 )'c/< such that \\y—b\\e2^ by
) -\\Az(y-b)\\2.
F(x,y) = (x,f (x,y))
for
every
(x,y)SRn x R m. Then, if |\x—a ||| \x—a \|2+| \e-Az(y-b)\ \ 2-e~^-K -1 \x-a\ \■\\e-Az(y-b)\ |. If we denote cr= ||jc—a\\ and quadratic
form
in
a,
y,
y= |\e-Az(y—b) ||, we remark that the
a 2—K*a •y-t-y2*^*”2^
is
positive
definite,
since
M ih ai Cristea
232
A=A'2-4-e-2/J 0
for
||x—a ||0. We finally proved that
( F (x ,y ) - F (a ,b ),H £( x - a y - b ) ) > 0 for every (x,y)eF rB s (a,b). As in Theorem 1, we prove that i(FrB^(ayb)y(ayb))= i(H £rB^(a, b)y(ayb)) = =;sign det//£=sign dctAz- ± lyand we obtain that F is a local homeomorphism on
UxV. Now, as in the classical proof of the implicite function theorem, we determine the implicite function (fk Let z = (a, b) be fixed and we d e n o te / (ayb^=c. Let 0 be such that there exists KEK((a, c)) such that F : B(a , 0 be such that B{ay/) x B (c, /) C V. If G : V-+ B(ayd)xB(b, 6), G = ( G 1, G 2) is the inverse of Fy then (xyz )—F(G(xyz)) = (G l (xyz)yf (G l (xyz),G 2 (xyz))) for every (XyZ)EVyhence it results that G l (xyz)=x and / (xyG 2(XyZ))=z for every (XyZ)EV. Let now r=min{ B{byc5) such that f (xy 0 and 0 and an unique continuous function ip : B{a,r) -» B(b,d) such that f (x, 3 and the conditions 1) and 2) hold on UxV\K , where K C U x V is closed and countable. Indeed, it is immediately that F is a light map and since F(K) is countable, it results from Theorem C that F is open on UxV. We apply now Theorem B. Theorem 7. Let UCRni VCRm be open and f : U x V -*Rm be differentiable. 00
Suppose that m + n>3 and there exists K C U x V such that K = U Kp with Kp closed P»\ sets and for every p& N and such that: 1) for every x ElU, the map fx : V ->Rm defined by fx(y)=zf (*,>>) for (x,y )E U x V is a
light map. 2) det-^ (2
)^ 0
for every z = (x>y)E.UxV\K. Then, for every z = (a ,b )& U x V there
exists r > 0 and 0 and an unique differentiable function
(x)) = f (a, b) for every xE.B(a, r). Proof. The map F(x,y) = (x,f (x,y)) defined on U x V is differentiable and we show tfiat it is a light map. Indeed, Lctz=(xiy)E;UxV. Since the map/^ is light, there exists U ^V iy )
such that f (x,w)^f (x,y)
for every wEFr Ui and /E/V.
Let
V-—B (x ,j)X U{ for iGN. We fix now /EjV and we show that F(xyy ) £ F (Fr V-). Indeed, otherwise
there
exists
(xi,yi)G ¥rV i
such
that
F(xifyi)=F(x,y)y
i.e.
(Xj>f(Xj>yi)) = (x>f(x,y))} which implies that x = x and f ix ,>»/) = / (a*,y). Now, since (*,>’y)EFr Viy it results that ^ E F r U>t and fx(yi) = fx(y)> Since this contradicts the choise of L/-, it results that F is a light map. Now, since F is differentiable, it results from Theorem F that m,n+n-.2(F(Kp))=0 for every /?E/V. Now, since detF for every (x, y) E Ux V \K, it results from Theorem E that/7 is a local homeomorphism UxV. The existence and the uniqueness of the function ip may be proved as in Theorem 4. Now, keeping the notations from Theorem 4, it results that the inverse of F, the map G = ( G 1, G2) is differentiable and the theorem is proved. 011
234
M ihai Cristea
Theorem 8. Let U ERn and VCRm be open and f : UxV-*Rm be differentiable. Suppose that there exists K C U x V such that mm+n(K)=0 and such that : 1) for every xCU} the map fx : V ->Rm defined by fx(y) —f (xyy) for (xyy )E U x V is a
light map. 2) det-^j* (z)>0 for every zEUxV\K or det*^ (z) 0 and c5>0 and an unique function (p : B(ayr)->B(b,d) such that
