LAPLACE EQUATION IN THE HALF-SPACE WITH A NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITION

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1 26 (2) MATHEMATICA BOHEMICA o. 2, LAPLACE EQUATIO I THE HALF-SPACE WITH A OHOMOGEEOUS DIRICHLET BOUDARY CODITIO Cherif Amrouche, Pau,Šárka ečasová, Praha Dedicated to Prof. J. ečas on the occasion of his 7th birthday Abstract. We dea with the Laace equation in the haf sace. The use of a secia famiy of weigted Soboev saces as a framework is at the heart of our aroach. A comete cass of existence, uniqueness and reguarity resuts is obtained for inhomogeneous Dirichet robem. Keywords: the Laace equation, weighted Soboev saces, the haf sace, existence, uniqueness, reguarity MSC 2 : 35J5, 58J. Introduction The urose of this aer is to sove the robem (P) u = f in R+, u = g on Γ = R, with the Dirichet boundary condition on Γ. The aroach is based on the use of a secia cass of weighted Soboev saces for describing the behavior at infinity. Many authors have studied the Laace equation in the whoe sace R or in an exterior domain. The main difference is due to the nature of the boundary and one of difficuties is to obtain the aroriate saces of traces. However, the haf-sace has a usefu symmetric roerty. Second author woud ike to thank the Grant Agency of the Czech Reubic o. 2/99/ 267 and both authors to the Barrande roject between the Czech Reubic and France. 265

2 Probem (P) has been investigated in weighted Soboev saces by severa authors, but ony in the Hibert cases ( = 2) and without the critica cases corresonding to the ogarithmic factor (cf. [2], [4]). We can aso mention the book by Simader, Sohr [5] where the Dirichet robem for the Laacian is investigated. Let Ω be an oen subset of R, 2. Let x =(x,...,x )beatyicaoint of R and r = x =(x x2 )/2. We use two basic weights: ϱ =(+r 2 ) /2 and g ϱ =n(2+r 2 ). As usua, D(R ) denotes the saces of indefinitey differentiabe functions with a comact suort and D (R ) denotes its dua sace, caed the sace of distributions. For any nonnegative integers n and m, rea numbers >, α and β, setting k = k(m,,, α) = we define the foowing sace: if + α/,...,m}, m α if + α,...,m}, (.) W α,β (Ω) = u D (Ω); λ k, ϱ α m+ λ (g ϱ) β D λ u L (Ω); k + λ m, ϱ α m+ λ (g ϱ) β D λ u L (Ω)}. In case β =,wesimydenotethesacebywα (Ω). ote that W α,β (Ω) is a refexive Banach sace equied with its natura norm [ α,β (Ω) = ϱ α m+ λ (g ϱ) β D λ u L (Ω) + λ k k+ λ m ϱ α m+ λ (g ϱ) β D λ u L (Ω)] /. We aso define the semi-norm ( /, u W α,β (Ω) = ϱ α (g ϱ) β D λ u L (Ω)) λ =m and for any integer q, wedenotebyp q the sace of oynomias in variabes of adegreesmaerthanorequatoq, with the convention that P q is reduced to } when q is negative. The weights defined in (.) are chosen so that the corresonding sace satisfies two roerties: (.2) D(R + ) is dense in Wα,β (R + ), and the foowing Poincaré-tye inequaity hods in W α,β (R + ). 266

3 Theorem.. Let α and β be two rea numbers and m an integer not satisfying simutaneousy (.3) + α,...,m} and (β ) =. a norm which is equiv- Then the semi-norm W α,β (R + ) defines on W α,β (R + )/P q aent to the quotient norm, (.4) u W α,β (R + ), α,β (R + )/P c u q W α,β (R + ) with q =inf(q, m ), whereq is the highest degree of the oynomias contained in W α (R + ), Proof. First, we construct a inear continuous extension oerator such that (.5) P : W α,β (R + ) W α,β (R ) satisfying (.6) Pu W α,β (R ) α,β (R + ). Since (.6) u W α,β (R ), α,β (R )/P q c u W α,β (R ) hods [cf. ], it automaticay imies the statement of our theorem. ow, we define the sace W α,β (R + )=D(R + ) W α,β (R + ) ; the dua sace of W α,β (R + ) is denoted by W α, β (R + ), where is the conjugate of : + =. Theorem.2. Under the assumtions of Theorem., the semi-norm W α,β (R + ) is a norm on W α,β (R + ) such that it is equivaent to the fu norm W α,β (R + ). We reca now some roerties of weighted Soboev saces W α,β (R + ). We have the agebraic and tooogica imbeddings W α,β (R + ) W m, α,β (R, + )... Wα m,β (R + ) 267

4 if + α/,...,m}. When + α = j,...,m}, thenwehave: W α,β (R + )... W m j+, α j+,β (R + ) W m j, α j,β (R + )... W, α m,β (R + ). ote that in the first case, the maing u ϱ γ u is an isomorhism from W α,β (R + ) onto W α γ,β (R + ) for any integer m. Moreover, in both cases and for any muti-index λ, the maing u W α,β (R + ) Dλ u W m λ, α,β (R + ) is continuous. Finay, it can be readiy checked that the highest degree q of the oynomias contained in W α,β (R + )isgivenby m ( q = + α) if + α,...,m} and (β ) + α j Z; j } and β [m ( + α)] otherwise, where [s] denotes the integer art of s. In the seque, for any integer q, we wi use the foowing oynomia saces: P q (Pq ) is the sace of oynomias (resectivey, harmonic oynomias) of degree q, P q is the subsace of oynomias in P q deending ony on the first variabes, x =(x,...,x ), A q (q ) is the subsace of oynomias Pq satisfying the condition (x, ) = (resectivey, x (x, ) = ) or equivaenty odd with resect to x (even with resect to x ), with the convention that P q,pq,p q,... are reduced to } when q is negative. 2. The saces of traces In order to define the traces of functions of W α,β (R + ), we introduce for any σ ], [ the sace (2.) W σ, (R )= u D (R ); w σ u L (R ), + } t σ dt u(x + te i ) u(x) dx<, R where 268 w = ϱ if σ, ϱ(g ϱ) /σ if = σ,

5 and e,...,e is a canonica basis of R. It is a refexive Banach sace equied with its natura norm ( σ, u (R ) = w σ + L (R ) which is equivaent to the norm i= ) / t σ dt u(x + te i ) u(x) dx R ( u R R w σ L (R + u(x) u(y) ) / ) x y +σ dx dy. For any s R +,weset } (2.2) W s, (R )= u W [s], [s] s (R ); λ =[s], D λ u W s [s], (R ). It is a refexive Banach sace equied with the norm s, (R ) = [s], [s] s (R ) + λ =s D λ u W s [s], (R ). We notice that this definition and the next one coincide with the definition in the first section when s = m is a nonnegative integer. For any s R + and α R, we then set } (2.3) Wα s, (R )= u W [s], [s]+α s (R ), λ =[s], ϱ α D λ u W s [s], (R ). Finay, for any integer m, we define the sace (2.4) X (R + )= u D (R + ); λ k, ϱ λ m (g ϱ ) D λ u L (R + ), } k + λ m, ϱ λ m D λ u L (R + ) with ϱ =(+ x 2 ) /2 and g ϱ =n(2+ x 2 ). It is a refexive Banach sace. We can rove that D(R + ) is dense in X (R + ). We observe that the functions from X (R + )andw (R + ) have the same traces on Γ = R (see beow). If u is a function, we denote its traces on Γ = R by x R, γ u(x )=u(x, ),...,γ j u(x )= j u (x, ). x j As in [3], we can rove the foowing trace emma: 269

6 Lemma 2.. For any integer m and rea number α, the maing γ : D(R + ) m j= D(R ) u (γ u,...,γ m u) can be extended by continuity to a inear and continuous maing sti denoted by γ from Wα m (R + ) to W m j, α (R ).Moreover,γ is onto and j= Ker γ = W α (R + ). 3. The Laace equation The aim of this section is to study the robem (P): (P) Theorem 3.. u = f in R+, u = g in Γ = R. Let be an integer and assume that (3.) /,...,} with the convention that this set is emty if =. For any f in W, (R + ) and g in W, (Γ) satisfying the comatibiity condition (3.2) ϕ A [+ ], f,ϕ ϕ = g, W, W, γ Γ where, Γ denotes the duaity between W, (Γ) and W, (Γ), robem(p) has a unique soution u W, (R + ) and there exists a constant C indeendent of u, f and g such that (3.3), (R + ) C( f W, (R + ) + g W, ). (Γ) Proof. First, the kerne of the oerator (,γ ): W, (R + ) W, (R + ) W, (Γ) 27

7 is recisey the sace A [+ / ] for any integer and A is reduced to } [+ ] when. Thanks to Lemma 2., et u g W, (R + ) be the ifting function of g such that u g = g on Γ and u g W, (R + ) C g W Then robem (P) is equivaent to,. (Γ) (3.4) v = f + ug in R +, v = on Γ. Set h = f + u g. For any ϕ W, (R )set It is cear that ϕ defined by ϕ(x,x )=ϕ(x,x ) ϕ(x, x ) if x >. ϕ W, W, (R + ). Then h can be extended to h π W, (R ) (R ), h π (ϕ) = h, ϕ W, (R, + ) W (R). + Moreover, h π W, (R ) = h W, (R ). + Let q be a oynomia in P[+ / ].Wecanwriteitintheform q = r + s, r A [+ / ] and s [+ /]. Then, h π,q = f + u g,r W, (R, + ) W (R + ) and aying the Green formua we get u g,r = u g rdx R + r = g, x W, (Γ) W, (Γ) (note that r =inr + and r =onγ). Thus,h π W, (R ) and it satisfies q P [+ / ], h π,q =. Reca that (cf. []) since (3.) hods, the oerators : W, (R ) W, P if, [+ ] : W, (R )/P [ ] W, (R ) P [ ] if = 27

8 are isomorhisms. Hence, there exists ṽ in W, (R ) such that ṽ = h π.owwe remark that the function w =, 2 ṽ beongs to W (R + )and w = h in R + and w = on Γ, i.e. w is a soution of (3.4). Remark. The kerne A [ + /] is reduced to } if andtop [ /] if =. With simiar arguments, we can rove the foowing theorem: (3.5) Theorem 3.2. Let be an integer and assume that /,..., }. Then for any f in W, (R + ) and g in W, (Γ), robem(p) has a unique soution u W, (R + )/A [+ /] and there exists a constant C indeendent of u, f and g such that inf u + q W, q A [+ (R + ) C( f W, (R + ) + g W ], ). (Γ) Theorem 3.3. assume that Let m be a nonnegative integer, et g beong to W + m (Γ) and (3.6) f W + m (R + ) if or m =, or (3.7) f Wm + (R + ) W, (R + ) if = and m. Then robem (P) has a unique soution u Wm + (R + ) and u satisfies m+, m (R + ) C( f Wm + (R + ) + g W + m ) if (Γ) or m = and m+, m (R + ) C( f W, (R + ) + f Wm + (R + ) + g ) W + m (Γ) if = and m. 272

9 Proof. First, we observe that for any integer m wehavetheincusion Wm + (R + ) W, (R + ) if orm =. Thus, under the assumtions (3.6) or (3.7) and thanks to Theorem 3., there exists a unique soution u W, (R + ) of robem (P). Let us rove by induction that (3.8) g W + m (Γ) and f satisfies (3.6) or (3.7) = u W m+, m (R + ). For m =, (3.8) is vaid. Assume that (3.8) is vaid for,,...,mand suose that g W +m+, m+ (Γ) and f W m+ (R + )with (a simiar argument can be used for f satisfying (3.7)). Let us rove that u W m+2, m+ (R + ). We observe first that W +m+, m+ (R + ) Wm m, (R + )andwm+ (Γ) W + m hence u beongs to Wm m+, (R + ) thanks to the induction hyothesis. ow, for i =,...,, (ϱ i u)=ϱ i f + 2 ( 2 ϱ r ( iu)+ ϱ + ) ϱ 3 i u. Thus, (ϱ i u) Wm m, (R + )andγ (ϱ i u) Wm m+, (R ). Aying the induction hyothesis, we can deduce that i u W m+, m+ (R + )fori =,...,. It remains to rove that v = u W m+, m+ (R + ). This is a consequence of the fact that v beongs to Wm (R + )and i u = i u W m+ (R + ), i =,...,, ( u)= u i 2 u W m+ (R + ). i= We can conude that u W m+2, m+ (R + ). Coroary 3.4. Let and m be two integers. (i) Under the assumtion (Γ), /,...,+}, 273

10 + for any f W m, m+ (R + ) and g Wm+ (Γ) satisfying the comatibiity condition (3.2) there exists a unique soution u W m+, m+ (R + ) of (P) and u satisfies m+, m+ (R + ) C( f W m, m+ (R + ) + g W + m+ (Γ) where C = C(m,,, ) is a constant indeendent of u, f and g. (ii) Under the assumtion ) m or /,..., m}, for any f W m, m (R + ) and g W + m (Γ) there exists a unique soution u W m+, m (R + )/A [+ /] of (P) and u satisfies inf q A [+ /] u + q W m+, m (R + ) C( f W m, m (R + ) + g W + m (Γ) ). References [] C. Amrouche, V. Giraut, J. Giroire: Weighted Soboev saces for Laace s equation in R. J. Math. Pures A. 73 (994), [2] T. Z. Boumezaoud: Esaces de Soboev avec oids our équation de Laace dans e demi-esace. C. R. Acad. Sci. Paris, Ser. I 328 (999), [3] B. Hanouzet: Esaces de Soboev avec oids. Aication au robème de Dirichet dans un demi-esace. Rend. Sem. Univ. Padova 46 (97), [4] V. G. Maz ya, B. A. Pamenevskii, L. I. Stuyais: The three-dimensiona robem of steady-state motion of a fuid with a free surface. Amer. Math. Soc. Trans. 23 (984), [5] C. C. Simader, H. Sohr: The Dirichet Probem for the Laacian in Bounded and Unbounded Domains. Pitman Research otes in Matehmatics Series 36, Addison Wesey Longman, Harow, Great Britain, 996. Authors addresses: Cherif Amrouche, Université de Pau et des Pays de L Adour, Laboratoire de Mathématiques Aiquées, I.P.R.A, Av. de Université, 64 Pau, France, e-mai: cherif.amrouche@univ-au.fr; Šárka ečasová, Mathematica Institute of the Academy of Sciences, Žitná 25, 5 67 Praha, Czech Reubic, e-mai: matus@math.cas.cz. 274