Kadlec, Jan

Kufner, Alois

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On the solution of the mixed problem (Preliminary communication)

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeJan Kadlec; Alois KufnerOn the solution of the mixed problem (Preliminary communication)Commentationes Mathematicae Universitatis Carolinae, Vol. 7 (1966), No. 1, 75--84Persistent URL: http://dml.cz/dmlcz/105041Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1966Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCommentatione8 Mathematicae Universitatie Carolinae 7, 1 (1965) ON THE SOLUTION OF THE .MIXED PROBLEM Jan KADLEC and Alois KUFNER, Praha (Preliminary communication) 1. Let H be a bounded domain in the plane £2> w h a*e boundary o)il fulfils locally a Lipsohltz condition. Decompose the boundary d-Q. into two parts, an . n, + p2 , where P^ has positive measure* Consider a function y on dSl such that (f> = 0 on H- , y> > 0 on r 2 • Let (1.1) A^^i^/a^rx^x^^J+ef^,^^ be an el l iptic differential operator of the second order, TV « C?V.n TVA ) the exterior normal vector to d£L and the exterior co-normal derivative. In this preliminary communication, we shall state so-me results concerning the solution of the mixed problem (1.2) Au m 4 in SL , (1.3) AJL + y ?~ * £ on a i l . - 75 -, It will: be point ad out that, under further assumptions, the solution may be sought in special weight spaces, with the weight function [djut rx , n; ) j * ; these make It possible to give a better characterization of the behavior of solutions in the neighborhood of those points on dil which are limit points of both f) and Pz * From this point of view it Is possible to solve the mixed problem also for those right-hand sides and boundary conditions for which the variational solution cannot be found without using weight functions (i.e. there exists no solution In the corresponding space with at m 0 ). Fur-thermore, one can (for various f and £. ) find better so-lutions than by the usual variational procedure. Remark: The fact that only the two-dimensional case is consi-dered, is not essential; in ia dimensions the difficulties are only in describing the position and shape of the parts H, and P2 of the boundary d ft . In thla section we shall Introduce some functional spaces. For simplicity we consider only real functions and functionala; derivatives are understood in the sense of dis-tribution-theory* The space of all functions AJL for which the norm (2.D U\f^ ^flu*'$ W 1^.*-" >* is finite will be denoted by W"} (SL) . - 76 -I»et a> f x ) *>e the distance between the point X *r ss(x &) and *"* ' and let ^ be a real number. It will be said that the function u, is in the space i~z ^(XL) if Denote by W/^ Cil ) the set of all functions with the finite norm Next, let V}*\ (XL) be the space of all functions* such that with the corresponding norm (2.3) ^ l ^ - ^ ^ . ^ * i ^ l L ^ , + lfejutt^)'4 • Obviously \£'J.Ol)c V ^ (Til) ; from the authors'results, [1], it follows that the function AJU e V™ fjfl ) has zero trace on P-j • It will be said that a function £ on dXL is in the space Witf£ (3-0. ) if there exists a function 9- 6 ^ ' J c ̂ ^ ̂  such that 9" is the trace of 9- on 3-0- . The function j^ is said to be the prolongation of 9- in XI , and we define 1 9 - V'<*>r<9ii> = w ' % V i "-> ' where the infimum i s taken over a l l prolongations ty of the function $. . Wa shal l always consider those prolongar tions ^ for which l$lwf%CA) * O tq,lwM)(aji) with c some posit ive constant, ' - 77 -The apace L% ^ ^ (V* ) ia defined as the aet of a l l functions AJU oa Vt With the finite norm (2.4) *">hit^cq) - f / f V - * ^ * • The moat important apace for our consideration ia the apaoa S&(SL) ~ \ £ (JU nLlt9t«(rg) with the norm 4/ (2.5) B^»s«, f A ) - C*<*ljw w + i ^ L ^ c v ) * • !,<*, lf**> z-ty i** The apaee L± ^ ( p^ ) eharaeterltee the trace of the function AJL 6 S/'i. (JL ) oa fl j the trace of *L oa Tt ia obfloualy aero aiaoe S£(fl) c V£*i Ol) . Remark. Let ac » 0 and y 6 0 2 c p f x ) ? where c ia a positive constant. Than It followa from the properties of tracea of functions from WM Cil.) that for efary M, e W9ci> (IL) (and thua X M also for e*ery >o^c l£l /(SL) ) the latter integral ia neceaaarily finite and can be estimated by the norm lAA,tv(4>(su . Thua in this case S™CJl)>- Vj^C-fL) • fa shall ao assume that g>(x)£c:p(x) . Lat SUCSt) be the aet of al l infinitely diffea?ea-tiafcle functions with compact support in i l .Let (X be a normal spmQ%t i . e . Q * 2J Til /* ia the norm of the apaoa (X » and let (X o S/*^ (ft ) algebraically and topologieally (for example (X » L% (ft ) ) . Let • f 8 -Q' %be the space of al l continuous linear functionals on Q ( i . e . Qf i s the space dual to Q )• The space dual to $c2 ^ (& ) i s denoted by %,--<. "̂̂ " ^ ; t n e si?*0© &u&l *° L^ ^ ^ C Pt ) may be identified with the space L2 _ ^ f Q ) in the usual manner. Finally, VV6" * OH) denotes the space dual Z j — oC to w/1*} Cdfl) . 3. Consider the operator A in the form (l»l) and the boundary value problem (1*2) and (1.3)» Assume that 1) the functions o^ • Cxi 9 xz ) are measurable bounded in H , and the quadratic form.X <*zjCx„x%)*ii w i s positive definite uniformly with respect to Xm(xirxt)eH; 2) the function cCx11xt) i s positive measurable; 3) the function <f 6c15 x2 ) (see boundary condition (1*2)) ful f i l s a Lips chit z condition; so we obviously have <fCx1fxz) & c pCx17xt ) for Cxifxz ) e d-& • To the operator *A there corresponds the bilinear form (3-D CU(A^,V-)~ fcu^Cx)§& §* dx+fc.yu.vdx ; from the e l l ipt ie i ty of A i t follows that \0,(AA,AA,)\ £ C t#>l$,w (XL} . To the mixed problem (1*2) and (1.3) there corresponds the bilinear form (3*2) B (AJL, V) - «, Ot, *r ) 4- y -*£2T d& q • 79 -defined on the carteaian product S2gC (St) x S2 ^ (SI) : i t can be eaaily shown that ( 3 . 3 ) \B(ufv)\ £ c, \\AC\\SCI) (Il) Wirllsci) (XL) > (3.4) \B(M.,«.n £ Cjlu.^„)(si) d.e. oc-0). Definition, The bilinear form E>(<ir, AA.) ±Q said to be \oc\-e l l i P t i c . i f there are positive constants c3 and c^ such that .. ^ujtv \&(AA,,<IT)\ £ C3 IACI ci) , AUjy \b(M.,nr)\ £ C^lvt ci) -3.,< Now, we have Theorem 1# Under the corresponding hypothese3 to the form & (AJU 7 <tr ) 7 there exists an interval J = (~ 3i i Tz ^ (ft > 0 ) such t h a t fo* oc e 3 the form 6 (<u*, or ) ia \dC I - e l l i p t i c . gfil§£i. If B (AMjtr) » B(ir-7 ^ ; ; then ^ = 3* ' Next we have, by the generalized Lax-Milgram theorem, t 2 j , the following Theorem^ 2 . Let aC e 7 P l e t F be a functional on the space Sf ^ (SI ) ( i . e . F e S / " ^ (SI ) ) . Then the-re exiats preci3ely one element ^ £ %7dC "̂̂ * ' euch ft(V-7 IA-; « F (v) Ci) tor every -v e £/_<*. (SL ) } and that , t < r , s /^r i i> * c* | F I I 5 / ^ w ' - 80 -4. From Theorem 2 we obtain the existence and uniqueness of the weak solution of the mixed problem (1.2) and (1.3); the exact formulation of this problem will be given in sec-tion 5. In all further considerations we assume oc e J , whe-re y is an interval as described by Theorem 1. Let f € Q,' and let r£. be a functional on the space of traces of functions from £ ^ - o t (SL ) \ we assu-4 me that £- can be decomposed thus: (4.1) 9- - £-, + &z + 9>9"s > ff/ ) where g^ e W2 ^ (d ft- ) (with the corresponding pro*-longation ^ e W_«_\ CH) ) , +_ e L ^ ^ C r_ ) and we put q._ = 0 for X e H, , and g. e WJ'^Cdn.) • for y & S^_ CSL) , ^^(yr) means the sam? as Let Y e Sf°_^ i-fl-) j setting (4.2) F(yJ - f Vif) - -ti'9,> ¥"} + f%^ + 9^ <>O > 1 we have the Theorem 3 . The funct ional F from (4.2) i s in the space I t follows from Theorem 2 that there i s p rec i se ly one element ur e S / ^ (Jl) such tha t B(ur,yr) = F(y ) fcr rvery -y e S^l ^ (St ) . Set AJL -» <w +- fyn f and l e t 9"1 ; 9 i ? ^3 ^ e funct ionals which form another decom-pos i t ion of the funct ional g. from ( 4 . 1 ) . i . e . - 81 -list " b e the functional corresponding to f , <fa t 9j x * and ^ i n a manner similar to (4 .2 ) , and l e t M> m <ur + + fa 1 where itf"* i s the solution of the equation B(W-* *y)m F*(ip) . Then we have Thaoggmi 4» Under the above hypotheses, AC* « 4*. • 5* .Definition, Iiet f be a functional.on S^;_^ fJl) ; let 9. be a functional on the space of traces of functions from $z -<*, (& ) with corresponding decomposition of the form (4.1)* Let F be defined by (4.2)# The function AA, € W/'* (XI ) is aaid to be a 2, «c weak solution of the mixed problem (1.2) and (ltlK if 1) .-U-& e S,% (XL) , 2) ft (M, - $< , r ) - F (r) for every f eS^ (H). Theorem 5. Let at e J . Then there exists, precisely one weak solution AM e W/4* (XI ) of the mixed problem (1*2) and (1.3), and the estimates # 1 At -<& asW & c IF 1 ^ , r j x ; and hold (the norms are considered in the corresponding spaces). Theorem 6« If Jscf i s suf f i c ient ly small and AM € Vz<c (XI) i s sueh that B (w , f ) * 0 for every V <s S/'L^ # * • ) , then .44, s& 0 . This theorem extends the assertion on uniqueness of solution, proved in theorem 5 for the apace Sf^ (X)- )f to the larger spaoe V/4[ (SI ) ' -82 -6. In this section it is established that the weak solution, defined la section 5, solves the problem (1»2) and (1.3) in the classical sense, if every element is a sufficiently smooth function* 1. The condition *L - fy € S>z(1)^ (SL } yields AJU «-2. We shall consider functions Y w h ich are zero la the neighbourhood of J"} ; the equality B Co, ~$j , if) =* Fty) % can then be rewritten as &Oc,y) » B C | ; , r ^ F(r)**a>(%<nr)+f^d6 + 4- fry) -a-(&,r)+J,±^Ed6i- f%rd(> = ='i(r)+f2§-de , i.e. 1 cC^^ + f^de^ffrdx + fM-de . By Green'a theorem, O,(<C,Y) =/AAI.Y dx+/§£ Yder > ' SA*.Гd*+f&Гd*-S**d*+f*=¥!Ldă i .e For y e 2)(J1) we have fAu, ,yc6c-=r/f y tff-X and thus ,. . -^ ̂  ^ AM*. ** T in TL • But in this ease we have for r̂ # 5 oa Q f3fcvde-f *g* r de and thue -§-£-.« -SL^i. oa Q ; therefore - 83 -establishing that our formulation i s meaningful. R e f e r e n c e s : tl] A. KUFNÊR, J. KADLEC: Characterization of functions with zero traces by integrals with weight functions, to appear in Ôasopis pro pëstovâ-ni matematiky, 1966. [z] J. NEÔAS: Sur une méthode pour résoudre l e s équations aux dérivées part ie l les du type e l l ipt ique , voisine de la variât ionel ie , Ann.Scuola Nor. Sup.Pisa,ser.3 ,16,4(1962),305-326. (Received December 13,1965) - 84 

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On the solution of the mixed problem (Preliminary communication)

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