Doubling property for the Laplacian and its applications (Course Chengdu 2007)

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1 Doubling popety fo the Laplacian and its applications Couse Chengdu 007) K.-D. PHUNG The oiginal appoach of N. Gaofalo and F.H. Lin Fo simplicity, we epoduce the poof of N. Gaofalo and F.H. Lin in the simplest case of the Laplacian. Let B yo,r = { y R N+, y y o < R }, with y o R N+ and R > 0. The unit sphee is defined by S N = B 0,. Theoem 4..- Let D R N+, N, be a connected bounded open set such that B 0, D. If v = v y) H D) is a solution of y v = 0 in D, then fo any 0 < M /, we have v y) dy N+ B exp ln 4) 0, v y) ) dy B 0,M B 0, v y) v y) dy. dσ B 0,M Poof.- We will use the following thee fomulas. Let R > 0, R f y) dy = f s) N ddσ s), 4.) B 0,R 0 S N d f y) dy = f Rs) R N dσ s) = f y) dσ y), 4.) dr B 0,R S N B 0,R B 0,R f y i y) dy = S N f Rs) s i R N dσ s). 4.3) The identity 4.) is the fomula of change of vaiable in spheical coodinate. 4.) comes fom 4.) when f is a continuous function. The identity 4.3), available when f is integable, taduces the Geen fomula. Indeed, B 0,R f y i y) dy = B 0,R f y) ν i y) dσ y) = S N f Rs) s i R N dσ s) because on B 0,R, ν x) = x R. Now, we will intoduce the following quantities. Let > 0, H ) = B 0, v y) dσ y), D ) = B 0, v y) dy, 4.4)

2 and N ) = D ) H ). 4.5) The goal is to show that N is a non-deceasing function fo 0 <. To this end, we will compute the deivatives of H and D. The computation of the deivative of H ) = S N v s) N dσ s) gives H ) = N N v s) dσ s) + v s) v) S N S N s s N dσ s) = N H ) + ) S v N s sn dσ s) = N H ) + div v ) N ddσ s) 0 S N = N H ) + B 0, v ) dy, but v ) = v v + v. Since y v = 0, one finds H ) H ) = N + B 0, v y) dy H ) Next, one emaks that indeed, B 0, yi v yi vdy Consequently, 4.6) becomes fom 4.7) H ) H ) = N ) v y) y dy = v y) v y) dσ y), 4.7) B 0, B 0, y = B 0, yi [v yi v] dy B 0, v y i vdy = B 0, [v yi v] y i y dσ y) because yv = 0 and on B 0,R, ν y) = y y. B 0, v y) v y) H ) y y dσ y) On anothe hand, the deivative of D ) = v) ρs 0 S ρ N dρdσ s) is N. 4.8) D ) = v) s S N dσ s) N = v) s S s s N dσ s) N = ) div 0 S v) s s N ddσ s) N ) = B 0, div v y dy, ) but div v y = v divy+ v ) y, with divy = N+. It emains to compute B 0, v ) ydy, when v = v y) H D). One have yj B 0, yi v ) ) y i dy = B 0, yj v y iy j vy i dy = [ ] B 0, yj yj v yi vy i dy B 0, y j v yi vy i dy B 0, yj v yi v yj y i dy = ) ) B 0, yj v yj y yi v y i y dσ y) B 0, yi v dy because y v = 0 and on B 0,R, ν y) = y y = y R. Consequently, D ) = N D ) + v B 0, y) y y dσ y), that is v D ) B D ) = N ) + 0, y) y y dσ y). 4.9) D )

3 Finally, the computation of the deivative of N ) = D) H) gives [ ] N ) = N ) + D ) D ) H ) H ) and one conclude fom 4.4), 4.8), 4.9) and 4.0) that y v N ) = N ) B0, y dy) ) B v 0, dy D ) H ), 4.0) v v y B0, ) y dy. Thanks to the Cauchy-Schwaz inequality, one deduce that N ) 0 i.e., N is non-deceasing on ]0, ]. Theefoe, fo, N ) N ) that is D) H) N ). Thus, fom 4.6) and 4.5), one have Consequently,, H ) H ) N N ). )) d H ) ln d N N ) d d ln ) By integating 4.) between R > 0 and R, one finds ) H R) ln H R) N N ) ln ), that is 0 < R /, S N v Rs) R) N dσ s) N e N ) ln 4. 4.) S N v Rs) R N dσ s). One conclude that fo any M /, B 0,M v y) dy = M v s) N ddσ s) 0 S N = M v Rs) R) N drdσ s) 0 S N N+ e N ) ln 4 M v Rs) R N drdσ s) 0 S N N+ e N ) ln 4 B 0,M v y) dy. Comment.- The above computations can be genealized to an elliptic opeato of second ode see [ GaL], [ Ku]). The impoved appoach of I. Kukavica It seems moe natual to conside the monotonicity popeties of the fequency function defined by B 0, v y) y ) dy B 0, v y) dy instead of B 0, v y) dy B 0, v y) dσ y). Following the ideas of Kukavica [ Ku], [ KN], see also [ AE]), one obtains the following thee lemmas. 3

4 . Monotonicity fomula We pesent the following lemmas. Lemma A.- Let D R N+, N, be a connected bounded open set such that B yo,r o D with y o D and R o > 0. If v = v y) H D) is a solution of y v = 0 in D, then Φ ) = B yo, v y) y y o ) dy B y o, v y) dy is non-deceasing on 0 < < R o, and d d ln v y) dy = N + + Φ )). B y o, Lemma B.- Let D R N+, N, be a connected bounded open set such that B yo,r o D with y o D and R o > 0. Let,, 3 be thee eal numbes such that 0 < < < 3 < R o. If v = v y) H D) is a solution of y v = 0 in D, then whee α = ln B y o, v y) dy ln + ln 3 ) ]0, [. B y o, v y) dy ) α B y o, 3 v y) dy ) α, The above two esults ae still available when we ae closed to a pat Γ of the bounday unde the homogeneous Diichlet bounday condition on Γ. Lemma C.- Let D R N+, N, be a connected bounded open set with bounday D. Let Γ be a non-empty Lipschitz open subset of D. Let o,,, 3, R o be five eal numbes such that 0 < < o < < 3 < R o. Suppose that y o D satisfies the following thee conditions: i). B yo, D is sta-shaped with espect to y o ]0, R o [, ii). B yo, D ]0, o [, iii). B yo, D Γ [ o, R o [. If v = v y) H D) is a solution of y v = 0 in D and v = 0 on Γ, then whee α = ln B yo, D ln + ln 3 ) α α v y) dy v y) dy v y) dy), B yo, B yo,3 D ) ]0, [.. Poof of Lemma B Let By applying Lemma A, we know that H ) = v y) dy. B yo, d d lnh ) = N + + Φ )). 4

5 Next, fom the monotonicity popety of Φ, one deduces that the following two inequalities ) ln H) H ) = N++Φ) d Consequently, ln ln ) H3 ) H ) and theefoe the desied estimate holds whee α = ln ln + ln 3 N + + Φ )) ln, = 3 N++Φ) d N + + Φ )) ln 3. ) ) H) H ) ln H3) H ) N + ) + Φ ln ), ln 3 ). H ) H )) α H 3 )) α,.3 Poof of Lemma A We intoduce the following two functions H and D fo 0 < < R o : H ) = B y o, v y) dy, D ) = B y o, v y) y y o ) dy. Fist, the deivative of H ) = v ρs + y 0 S N o ) ρ N dρdσ s) is H ) = B v y) y dσ y). Next, o, ecall the Geen fomula v ν Gdσ y) ν v ) Gdσ y) = v Gdy v ) Gdy. B y o, B y o, B y o, B y o, We apply it with G y) = y y o whee G Byo, = 0, ν G Byo, =, and G = N + ). It gives H ) = B N + ) v y dy + o, B yo, v ) y y o ) dy = N+ H ) + B yo, div v v) y y o ) dy ) = N+ H ) + v + v v y y o ) dy. Consequently, when y v = 0, that is H ) H) = N+ + D) H) B yo, H ) = N + H ) + D ), the second equality in Lemma A. A.) Now, we compute the deivative of D ) = v) ρs+yo 0 S ρ ) ρ N dρdσ s): N D ) = d d ) v) ρs+yo 0 S ρ N dρdσ s) N = v) ρs+yo 0 S ρ N dρdσ s) N = B yo, v dy. S N v) s+yo N dσ s) A.) 5

6 Hee, we emak that B v y dy = N+ o, D ) + 4 B y y y o) v dy o, B yo, v y y o ) v y y o ) dy, indeed, A.3) N + ) B v y y y o ) dy o, = B yo, div v y y o ) ) y y o ) dy B yo, v y y o )) y y o ) dy = B y y i v y y o )) y i y oi ) dy because on B yo,, = y y o o, = B v y y i v y y o ) y i y oi ) dy o, B v y y i y oi )) y i y oi ) dy o, = B v y y i v y y o ) y i y oi ) dy + o, B v y y y o dy, o, Theefoe, and B yo, yj v y i y j v = B y o, y j y i y oi ) yj v yi v + B y o, y j y i y oi ) yj v yi v + B yo, y i y oi ) y j v yi v y y o ) y i y oi ) dy y y o )) dy y y o ) dy y y o ) dy + B yo, y i y oi ) yj v yi v yj y y o ) dy = 0 because on B yo,, = y y o + B yo, v y y o ) dy + B y y y o) v v y y o ) dy o, B y o, y y o) v dy. N + ) B v y y y o ) dy = o, B v y dy 4 o, B y y y o) v dy o, + B y y y o) v v y y o ) dy, o, and this is the desied estimate A.3). Consequently, fom A.) and A.3), we obtain, when y v = 0, the following fomula D ) = N + D ) + 4 y y o ) v dy. B yo, A.4) The computation of the deivative of Φ ) = D) H) gives Φ ) = H ) [D ) H ) D ) H )], which implies using A.) and A.4) that [ ] H ) Φ N+ ) = D ) + 4 B y y y o) v dy H ) [ N+ o, H ) + D )] D ) 4 ) B yo, y y o ) v dyh ) D ) 0, = indeed, thanks to an integation by pats and using Cauchy-Schwaz inequality, we have D ) = 4 B v v y y y o) dy o, ) 4 B y y yo, o) v dy) B v yo, dy ) 4 B y y yo, o) v dy H ). ) 6

7 Theefoe, we have poved the desied monotonicity fo Φ and this completes the poof of Lemma A..4 Poof of Lemma C Unde the assumption B yo, D Γ fo any [ o, R o ), we extend v by zeo in B yo,r o \D and denote by v its extension. Since v = 0 on Γ, we have v = v D in B yo,r o, v = 0 on B yo,r o D, v = v D in B yo,r o. Now, we denote = B yo, D, when 0 < < R o. In paticula, = B yo,, when 0 < < o. We intoduce the following thee functions: and H ) = v y) dy, D ) = v y) y y o ) dy, Φ ) = D ) H ) 0. Ou goal is to show that Φ is a non-deceasing function. Indeed, we will pove that the following equality holds d d lnh ) = N + ) d d ln + Φ ). C.) Theefoe, fom the monotonicity of Φ, we will deduce that in a simila way than in the poof of Lemma A) that ) ) ln H) H ) ln H3) H ) ln N + ) + Φ ) ln, 3 and this will imply the desied estimate whee α = ln v y) dy ln + ln 3 ). B y o, v y) dy ) α 3 v y) dy ) α, Fist, we compute the deivative of H ) = B yo, v y) dy = v ρs + y 0 S N o ) ρ N dρdσ s). H ) = v s + y S N o ) N dσ s) = v s + y S N o ) s s N dσ s) ) = B yo, div v y) y y o ) dy ) = B y N + ) v y) + v y) y y o ) dy o, = N+ H ) + v y) v y) y y o ) dy. C.) Next, when y v = 0 in D and v Γ = 0, we emak that D ) = v y) v y) y y o ) dy, C.3) 7

8 indeed, v y y o ) dy = div v v y y o )) dy vdiv v = v v y y o ) dy v v because on B yo,, = y y o and v Γ = 0 = v v y y o ) dy because y v = 0 in D. Consequently, fom C.) and C.3), we obtain y y o )) dy y y o ) dy H ) = N + H ) + D ), C.4) and this is C.). On anothe hand, the deivative of D ) = v) ρs+yo 0 S ρ ) ρ N dρdσ s) is N D ) = v) ρs+yo 0 S ρ N dρdσ s) N = v y) dy. C.5) Hee, when y v = 0 in D and v Γ = 0, we will emak that v y) dy = N+ D ) + 4 B y y y o) v y) dy o, + Γ B y νv y y o ) y y o ) νdσ y), o, C.6) indeed, N + ) v y y o ) dy = div v y y o ) ) y y o ) dy v y y o )) y y o ) dy = Γ B yo, v y y o ) y y o ) νdσ y) yi v y y o )) y i y oi ) dy = Γ B yo, v y y o ) y y o ) νdσ y) v yi v y y o ) y i y oi ) dy + v y y o dy, and yj v y i y j v Theefoe, when y v = 0 in D, we have = yj y i y oi ) yj v yi v + yj y i y oi ) yj v yi v + y i y oi ) y j v yi v y y o ) y i y oi ) dy y y o )) dy y y o ) dy y y o ) dy + y i y oi ) yj v yi v yj y y o ) dy = Γ B ν y j y i y oi ) yj v yi v y y o )) dσ y) o, + v y y o ) dy +0 because y v = 0 in D y y o ) v dy. N + ) v y y o ) dy = Γ B v y y y o ) y y o ) νdσ y) o, Γ B y y j vν j y i y oi ) yi v) y y o ) dσ y) o, + u dy 4 y y o ) v dy. 8

9 By using the fact that v Γ = 0, we get v = v ν) ν on Γ and we deduce that N + ) v y y o ) dy = Γ B yo, ν v y y o ) y y o ) νdσ y) and this is C.6). + v dy 4 y y o ) v dy, Consequently, fom C.5) and C.6), when y v = 0 in D and v Γ = 0, we have D ) = N + D )+ 4 y y o ) v y) dy+ ν v y y o ) y y o ) νdσ y). Γ B yo, C.7) The computation of the deivative of Φ ) = D) H) gives Φ ) = H ) [D ) H ) D ) H )], which implies fom C.4) and C.7), that H ) Φ ) = 4 ) y y o ) v y) dy H ) D ) + Γ B yo, ν v y y o ) y y o ) νdσ y) H ). Thanks to C.3) and Cauchy-Schwaz inequality, we obtain that 0 4 y y o ) v y) dy H ) D ). the inequality 0 y y o ) ν on Γ hols when B yo, D is sta-shaped with espect to y o fo any ]0, R o [. Theefoe, we get the desied monotonicity fo Φ which completes the poof of Lemma C. 3 Quantitative unique continuation popety fo the Laplacian Let D R N+, N, be a connected bounded open set with bounday D. Let Γ be a non-empty Lipschitz open pat of D. We conside the Laplacian in D, with a homogeneous Diichlet bounday condition on Γ : y v = 0 in D, v = 0 on Γ, 4.) v = v y) H D). The goal of this section is to descibe intepolation inequalities associated to solutions v of 4.). Theoem 4..- Let ω be a non-empty open subset of D. Then, fo any D D such that D D Γ and D \Γ D ) D, thee exist C > 0 and ]0, [ such that fo any v solution of 4.), we have ) v y) dy C v y) dy v y) dy). D ω D O in a equivalent way, 9

10 Theoem Let ω be a non-empty open subset of D. Then, fo any D D such that D D Γ and D \Γ D ) D, thee exist C > 0 and ]0, [ such that fo any v solution of 4.), we have D v y) dy C ) v y) dy + ε v y) dy ε > 0. ε ω D Poof of Theoem 4..- We divide the poof into two steps. Step.- We apply Lemma B, and use a standad agument voi e.g., [ Ro]) which consists to constuct a sequence of balls chained along a cuve. Moe pecisely, we claim that fo any non-empty compact sets in D, K and K, such that meask ) > 0, thee exists ]0, [ such that fo any v = v y) H D), solution of y v = 0 in D, we have ) v y) dy v y) dy v y) dy). 4.3) K K D Indeed, let δ > 0 and q j R N fo j = 0,,, m, one can constuct a sequence of balls { } B qj,δ j=0,..,m, such that the following inclusion hold K B q0,δ K B qm,δ o fo some δ o > 0 B qj+,δ B qj,δ j = 0,.., m B qj,3δ D j = 0,.., m. Then, thanks to Lemma B, thee exist α, α, ]0, [, such that K v y) dy B q v y) dy m,δo B q m,δ v y) dy) α B q m,3δ v y) dy B qm,δ v y) dy) α D v y) dy ) α ) α B qm,δ v y) dy) α D v y) dy) α ) α D v y) dy which implies the desied inequality 4.3). B q0,δ v y) dy) D v y) dy), Step.- We apply Lemma C, and choose y o in a neighbohood of the pat Γ such that the conditions i, ii, iii, hold. Next, by an adequate patition of D, we deduce fom 4.3) that fo any D D such that D D Γ and D \Γ D ) D, thee exist C > 0 and ]0, [ such that fo any v = v y) H D) such that y v = 0 on D and v = 0 on Γ, we have This completes the poof. ) v y) dy C v y) dy v y) dy). D ω D ) α Remak.- Fom standad minimization technique, the above inequality implies D v y) dy C ) v y) dy + ε v y) dy ε > 0. ε ω D 0

11 Indeed, we denote A = D v y) dy 0, B = ω v y) dy and E = D v y) dy. We know that thee exist C > 0 and ]0, [ such that A CB E. Theefoe, A C B E A ). Now, if E A ε, then A C B ) ε. And, if E A > ε, then A εe. Consequently, one obtain the desied intepolation inequality. Convesely, suppose that thee exist C > 0 and ]0, [ such that D v y) dy C A then, we choose ε = E in ode to get A CB E. ) v y) dy + ε v y) dy ε > 0, ε ω D Comment.- The above computations can be genealized to solutions of the following elliptic system y u = f in D, u = 0 on Γ, u = u y) H D), y ) f H H0 D), in ode to get the following estimate D u y) dy C y ) ) f + u y) dy u y) dy). L D) ω D 4 Quantitative unique continuation popety fo the elliptic opeato t + In this section, we pesent the following esult to be compaed to [ LeR]). Theoem Let be a Lipschitz connected bounded open set of R N, N. We choose T > 0 and δ ]0, T/[. We conside the elliptic opeato of second ode in ]0, T [ with a homogeneous Diichlet bounday condition on 0, T ), t u + u = 0 on ]0, T [, u = 0 in ]0, T [, 4.4) u = u x, t) H ]0, T [). Then, fo any ϕ C0 0, T )), ϕ 0, thee exist C > 0 and ]0, [ such that fo any u solution of 4.4), we have T δ δ T u x, t) dxdt C 0 ) T u x, t) dxdt 0 ϕu x, t) dxdt). Poof.- We apply Theoem 4. with D = ]0, T [, ]δ, T δ[ D, y = x, t), y = t +.

12 5 Quantitative unique continuation popety fo the sum of eigenfunctions The goal of this section is to obtain the following esults to be compaed to [ LZ] o [ JL]). Theoem Let be a bounded open set in R N, N, eithe convex o C and connected. Let ω be a non-empty open subset in. Then, thee exists C > 0 such that fo any sequence {a j } j of eal numbes and any intege M >, we have a j Ce C λ M ω a j e j x) dx, whee {λ j } j and {e j } j ae the eigenvalues and eigenfunctions of in H 0 ), constituting an othonomal basis in L ). O in an equivalent way, Theoem Let be a bounded open set in R N, N, eithe convex o C and connected. Let ω be a non-empty open subset in. Then, thee exists C > 0 such that fo any sequence {a j } j of eal numbes and any R > λ, we have {j;λ j R} a j Ce C R ω {j;λ j R} a j e j x) dx, whee {λ j } j and {e j } j ae the eigenvalues and eigenfunctions of in H 0 ), constituting an othonomal basis in L ). Poof of Theoem We divide the poof into thee steps. Step.- Fo any a j R, we intoduce the solution w x, t) = λj ) a j e j x) ch t χ x) a j e j x), whee χ C 0 ω), χ = in ω ω. Recall that cht = e t + e t ) /. Theefoe, w solves t w + w = f in ]0, T [, w = 0 on ]0, T [, w = t w = 0 on ω {0}, w = w x, t) H ]0, T [), fo any T > 0, whee f = χ a j e j H 0 ). We denote by w the extension of w by zeo in ω ] T, 0[. Theefoe, w solves t w + w = f 0,T ) in ]0, T [ ω ] T, 0[, w = 0 in ω ] T, 0[, w = 0 on ]0, T [. At pesent, we define D, a connected open set in R N+, satisfying the following six conditions: i). ]δ, T δ[ D fo some δ ]0, T/[ ; ii). ]δ, T δ[ D ; iii). D ]0, T [ ω ] T, 0[ ;

13 iv). thee exists a non-empty open subset ω o D ω ] T o, 0[ fo some T o ]0, T [ ; v). D C if is C and connected ; vi). D is convex with an adequate choice of δ, T o ) if is convex. In paticula, w H D). Step.- We claim that thee exists g H ] T, T [) H0 ] T, T [) H D) such that { t g + g = f 0,T ) in ] T, T [, g = 0 on ] T, T [), and g L D) f L ]0,T [). 4.5) Indeed, we will poceed with six substeps when is C and connected the case whee is convex is well-known since then ] T, T [ is convex). We denote h = f 0,T ) L ] T, T [). Substep : one ecall that h L ] T, T [) implies the existence of g H 0 ] T, T [). Substep : thanks to the inteio egulaity fo elliptic systems, fo any D 0 ] T, T [, g H D 0 ). Substep 3: thank to the bounday egulaity fo elliptic systems, but not closed to the bounday { T, T }, g is also locally in H because is C. Substep 4: we extend the solution at t = T as follows. Let h x, t) = h x, t) fo x, t) ] T, T [ and h x, t) = h x, T t) fo x, t) ]T, 3T [. Thus h L ] T, 3T [). Let g x, t) = g x, t) fo x, t) [ T, T [ and g x, t) = g x, T t) fo x, t) [T, 3T ]. Thus, g solves { t g + g = h in ] T, 3T [, g = 0 on ] T, 3T [). By applying the bounday egulaity as in substep 3, one obtain that g H ]0, T [). In paticula, g H ]0, T [). Substep 5: we extend in a simila way at t = T in ode to conclude that g H ] T, 0[). Substep 6: finally, we multiply t g + g = h by ) g and integate by pats ove ] T, T [, in ode to obtain T T tg H ) dt + g L ] T,T [) which gives the desied inequality 4.5). = T 0 f x) ) g x, t) dxdt = T 0 f x) ) ) g x, t) dxdt because f H0 ) f L ]0,T [) g L ] T,T [) fom Cauchy-Schwaz. Step 3.- Finally, we apply Theoem 4.3 with y = t +, v = w g in D with Γ = ]0, T [ and ]δ, T δ[ D D such that D D Γ and D \Γ D ) D in ode that which implies that D w g dy C D w dy C ) w g dy + ε w g dy ε > 0, ε ωo D ) g dy + ε w dy ε ]0, [, ε D D whee we have used that w = 0 in ω o. Fom 4.5), we conclude that thee exist C > 0 and ]0, [ such that T δ δ w x, t) dxdt C ) T T f x) dxdt + ε w x, t) dxdt ε > 0. ε 0 0 3

14 On anothe hand, we have the following inequalities T 0 f x) dxdt = T 0 χ x) a j e j x) dxdt T ω χ x) a j e j x) dx, T δ δ T 0 w x, t) dxdt = T 0 a j e j x) ch λ j t ) χ x) a j e j x) dxdt T e λ M T a j + T ω χ x) a j e j x) dx, t) λj a j e j x) ch dxdt T δ δ w x, t) dxdt + T Consequently, fom the last fou inequalities, we deduce that fo any ε > 0, T δ) a j T δ δ a j e j x) ch λ j t ) dxdt C ) ε T ω χ x) a j e j x) dx +4ε T e λ M T a j + T ω χ x) a j e j x) dx +T ω χ x) a j e j x) dx. Choosing ε = 8 T δ) T e λ M T, we obtain the existence of C > 0 such that a j Ce C λ M ω a j e j x) dx. ω χ x) a j e j x) dx. 6 Application to the wave equation Fom the idea of L. Robbiano which consists to use an intepolation inequality of Hölde type fo the elliptic opeato t + and the FBI tansfom intoduced by G. Lebeau et L. Robbiano, we obtain the following estimate of logaithmic type. Theoem Let be a bounded open set in R N, N, eithe convex o C and connected. Let ω be a non-empty open subset in. Then, fo any β ]0, [, thee exist C > 0 and T > 0 such that fo any solution u of t u u = 0 in ]0, T [, u = 0 on ]0, T [, u, t u), 0) = u 0, u ), 4

15 with non-identically zeo initial data u 0, u ) H 0 ) L ), we have u 0, u ) H 0 ) L ) e C u 0,u! ) /β H 0 ) L ) u 0,u ) L ) H ) u L ω ]0,T [). 7 Application to the heat equation In this section, we popose the following esult. Theoem Let be a bounded open set in R N, N, eithe convex o C and connected. Let ω be a non-empty open subset in. Then, fo any T > 0, thee exists C > 0 such that fo any u solution of t u u = 0 in ]0, T [, u = 0 on ]0, T [,.3) u, 0) = u o, with non-identically zeo initial data u o H0 ), and fo any t o ]0, T [, we have 0 uo C@ u o L ) Ce to +t H 0 ) o A uo L ) u, t o ) L ω). Comment.- We also have that u o H ) Ce C uo to +t L ) o uo H )! u, t o ) L ω). Remak.- The quantitative unique continuation popety fom ω {t o } fo paabolic opeato with space-time coefficients was established by L. Escauiaza, F.J. Fenandez and S. Vessella [ EFV]). Poof of Theoem We decompose the poof into two steps. Fist, in step, we will pove that the solution u of.3) satisfies the following estimate ) u o L ) exp t o u o H 0 ) u o L ) u, t o ) L ). Next, in step, we will pove that the solution u of.3) satisfies the following estimate ) u x, t o ) dx C e C to u x, 0) dx u x, t o ) dx). ω Finally, the above inequalities imply the existence of C > 0 such that u o L ) C C/ u o ) H e 0 to exp t ) o u o u x, t o ) dx. L ) ω Poof of step.- Let us intoduce fo almost t [0, T ] such that the solution of.3) satisfies u, t) 0, the following quantity. Φ t) = u x, t) H 0 ) u x, t) L ). 5

16 We begin to check that Φ is a non-inceasing function on [0, T ]. This monotonicity popety holds because fo any initial data in a dense set of H0 ), we have that d dtφ t) 0. Indeed, fom the following two equalities { d d dt u x, t) L ) + u x, t) H o ) = 0, dt u x, t) H 0 ) + u x, t) L ) = 0, we can deduce that d dt Φ t) = u x, t) 4 L ) [ u x, t) L ) u x, t) L ) + u x, t) 4H 0 ) ]. Theefoe, we get by classical density agument and Cauchy-Schwaz inequality that fo any solution u of.3), u, t) 0 a.e., and any t [0, T ], Φ t) Φ 0). On anothe hand, we also have that d dt u x, t) L ) + Φ t) u x, t) L ) = 0, which implies that 0 d dt u x, t) L ) + Φ 0) u x, t) L ). Theefoe, by Gonwall Lemma, we get the desied estimate Poof of step.- Let λ,λ, and e,e, be the eigenvalues and eigenfunctions of in H0 ), constituting an othonomal basis in L ). Fo any u o = u, 0) = α j e j in L ) whee j α j = u oe j dx, the solution u of.3), can be witten u x, t) = α j e j x) e λjt. Let t o ]0, T [. We intoduce see [ Lin] o [ CRV]) the solution w x, t) = λj ) α j e j x) e λjto ch t χ x) α j e j x) e λjto, j j j whee χ C 0 ω), χ = in ω ω. Theefoe, w solves t w + w = f in ]0, T [, w = 0 on ]0, T [, w = t w = 0 on ω {0}, w = w x, t) H ]0, T [), whee f = χu, t o ) H 0 ). Consequently, in a simila way than in the poof of Theoem 4.5, fo any δ ]0, T/[, thee exist C > 0 and ]0, [ such that we have T δ δ w x, t) dxdt C On anothe hand, the following inequalities hold. T f x) dxdt T T 0 0 w x, t) dxdt T j ) T T f x) dxdt + ε w x, t) dxdt ε > 0 ε 0 0 ω χ x) u x, t o ) dx, α je λ jt o λ j T) + T ω χ x) u x, t o ) dx, αj e λ jt o λ j T) = α j {j ; λ j T } j e λjto λj T) + α {j ; λ to j> T } j e λjto λj T) to e T to αj, j 6

17 T δ δ t) λj α j e j x) e λ jt o ch dxdt j T δ δ w x, t) dxdt+t Consequently, fom the fast five inequalities, we deduce that fo any ε > 0, T δ) αj e λjto T δ) αj e λ jt o λ j δ) j j T δ δ α j e j x) e λjto ch λ j t ) dxdt T C ε +4T ε j ) Finally, thee exists C > 0 such that fo any t o > 0, u x, t o) dx = αj e λjto j C ε which implies the desied estimate, u x, t o ) dx C e C ) to ω χ x) u x, t o) dx αj + ) ω χ x) u x, t o) dx e T to j +T ω χ x) u x, t o) dx. ω χ x) u x, t o ) dx, ) ω u x, t o) dx + εe C to u x, 0) dx ε ]0, [, ) u x, 0) dx u x, t o ) dxdt). ω 8 Notes on the papes in efeence... to be completed. some efeence to papes of Alessandini have to be add too...) Refeences [ A] S. Angenent, The zeo set of a paabolic equation, J. eine angew. Math ), [ AE] V. Adolfsson and L. Escauiaza, C,α domains and unique continuation at the bounday, Comm. Pue Appl. Math., 50, 3 997) [ CRV] B. Canuto, E. Rosset and S. Vessella, Quantitative estimates of unique continuation fo paabolic equations and invese initial-bounday value poblems with unknows boundaies, Tans. Ame. Math. Soc ), [ E] L. Escauiaza, Communication to E. Zuazua. [ EFV] L. Escauiaza, F.J. Fenandez and S. Vessella, Doubling popeties of caloic functions, Appl. Anal. 85, ) [ GaL] N. Gaofalo and F H. Lin, Monotonicity popeties of vaiational integals, A p -weights and unique continuation, Indiana Univ. Math. J. 35, 986)

18 [ JL] D. Jeison and G. Lebeau, Nodal sets of sum of eigenfunctions, Hamonic analysis and patial diffeential equations Chicago, IL, 996), Univ. Chicago Pess, Illinois, 999) [ Ku] I. Kukavica, Level sets fo the stationay Ginzbug-Landau equation, Calc. Va ), 5-5. [ Ku] I. Kukavica, Quantitative uniqueness fo second ode elliptic opeatos, Duke Math. J ) [ KN] I. Kukavica and K. Nystöm, unique continuation on the bounday fo Dini domains, Poc. of Ame. Math. Soc. 6, 998) [ Lin] F. H. Lin, A uniqueness theoem fo the paabolic equations, Comm. Pue Appl. Math ), [ LeR] G. Lebeau and L. Robbiano, Contôle exacte de l équation de la chaleu, Comm. Pat. Diff. Eq ) [ LeR] G. Lebeau and L. Robbiano, Stabilisation de l équation des ondes pa le bod, Duke Math. J. 86, 3 997) [ LZ] G. Lebeau and E. Zuazua, Null-contollability of a system of linea themoelasticity, Ach. Mech. Anal. 4, 4 998) [ M] L. Mille, Unique continuation estimates fo the laplacian and the heat equation on non-compact manifolds, Math. Res. Lettes, 005) [ Ro] L. Robbiano, Théoème d unicité adapté au contôle des solutions des poblèmes hypeboliques, Comm. Pat. Diff. Eq. 6 99) [ Ro] L. Robbiano, Fonction de coût et contôle des solutions des équations hypeboliques, Asymptotic Analysis, 0 995)