Well-Posedness of Difference Elliptic Equation

The exact with respect to step h∈(0,1] coercive inequality for solutions in Ch of difference elliptic equation is established.


INTRODUCTION
It is well-known in the theory of differential equations that the coercive inequalities approach appeared to be very useful for the investigation of general boundary value problems for elliptic and parabolic differential equations.
The coercive inequalities hold also for various difference analogues of such problems.These in- equalities, evidently, permit to prove not only the existence of solutions but also well-posedness of these problems.Main role of the coercive in- equalities for difference problems lies in that they present a special type of stability, which permits the existence of exact, i.e. two-sided estimates of the rate of convergence approximate solutions (with respect to the corresponding coercive norms).
As it turns out, there are situations when the difference problems are well-posed, but their lim- it variants-differential problemsare ill-posed.This paper deals with a consideration just of one of such cases.The established here exact (with respect to step h of difference scheme) coercive 219 inequality gives the possibility to find (almost) exact estimates of the rate of convergence of approximate solutions in the case when the differ- ential problem is ill-posed.

lll-Posedness of Differential Elliptic
Equation in C We will consider the simplest elliptic differential equation (0.1) on the plane R 2 of points x-(x1, x2).It is natu- ral to call function v(x) v(x,xz) the (classical) solution of Eq. (0.1), if it has the continuous and bounded partial derivatives till the second order, and if it satisfies Eq. (0.1).We will consider dif- ferential equation (0.1) as the operator equation in the Banach space C-C(R2) of continuous and bounded (scalar) functions (x)= (x,x2) with norm I1 1 c sup (0.2) xcR For the existence of such solution v of Eq. (0.1), evidently, it is necessary that fc C. (0.3) We will say that Eq. (0.1) is well-posed in C (see [1]), if the following two conditions are fulfilled: (a) There exists the unique solution v(x)= v(x;f) in C of Eq. (0.1) for any fc C. It means, in particular, that formula Iv(f)](x) v(x ;f) (0.4) defines the homogeneous and additive operator, acting from C in the Banach space C 2 of (scalar) functions b(x) b(x,x2), having continuous and bounded partial derivatives till the second order, with norm + IIo2/Ox, Oxalic. (0.5) i,j leads us to coercive inequality Ilvll2 M. Ilfllc (0.9) for solutions in C of Eq. (0.1) with some _< M< +cx, does not depend on f C. How- ever, it is well-known (see [1]) that Eq. (0.1) is not well-posed in C. The corresponding counter- example can be given by (0 < c < 1) It means that vc e C and for 0 < Xl 2 + x <_ 1/9 + a,l ( (0.11) This property is, evidently, equivalent to in- equality IIv(f)llc _< M. Ilfllc (0.6) for some continuous functions a,i(x) a,i(x, X2) (i 1,2).Therefore, evidently, Eq. (0.1) is ill- posed in C. It means that coercive inequality (0.9) is not true for any solution in C of Eq. (0.1).
(0.19) Since, evidently, operators (0.16) are bounded (for fixed h), then, in virtue of contraction map- ping principle, Eq. (0.18) for any fh C h has the unique solution vhE Ch, if A>0 is sufficiently large.Further we apply the maximum principle (to system (0.19)) and obtain a priori estimate is true.Since Dk h'2 (k-1,2) are bounded operators (for fixed h), then coercive inequality (0.17) is true.The value Mc(h) in this inequality, evidently, must tend to / , when h / 0, since the differential coercive inequality (0.9) is not true.It is the consequence of ill-posedness in C of differential equation (0.1).From estimate (0.21) and from formulas (0.16), evidently, it follows that we can put Mc(h) M. h -2 (0.22) in inequality (0.17) for some <_ M < + ec, does not depend on fh C h and 0 < h <_ 1.It turns out that essentially more exact result is true.Namely, for solution v h in C h of Eq. (0.13) coercive inequality (0.17) takes place for mc(h) mo ln /h (O < h 1 / 2 (0.23) with some _< M0 < +c, does not depend on fh and h.It is, in particular, the consequence of theory of difference equations which is devoted in this paper.Formula (0.23) means that sup c fhch.fh=O _< Mo.In 1lb.
0.3.The Almost Exact Estimate of Convergence Rate Let v be the solution in C of Eq. (0.1), having the continuous and bounded partial derivatives till the fourth order.Let further vi,j (i,j--oc, +oc) be the solution of system (0.12) for fi,j f ih,jh (0.30) Then, evidently, values Zi,j v(ih,jh) Fi,j (i,j--oc, +oc) (0.31) are the solutions of the system Zi,j [(Zi+ 1,j 2. Zi,j / Zi_ 1,j)h -2 Jr-(zi,j-t-2.zi,j Jr-zi,j- (0.32) and for values F i,j estimates Iri,j{ M. h 2 (0.33) take place for some <_M< +oo, does not depend on h.Therefore, from (0.24) it follows that estimate from above is true for some _< M1 < + oc, does not depend on h.
Finally, let f(xl,x2)O be the smooth func- tion, which partial derivatives till the second or- der sufficiently quickly tend to zero, when x 2 + x22 tend to infinity.Then, evidently, sup lO4v/Ox / 04v/Ox24] > 0, (0.35) is true for some 0 < m < / oc, does not depend on h.Estimate (0.36) and triangle inequality lead us to estimate from below nll .
for some 0 < ml < /oc, does not depend on h.Estimates (0.34) and (0.37) give the almost exact estimate of convergence rate of difference method (0.12) in the difference coercive norm.
is established for its solutions v a in Ca'a(E) with some _< M < oc, does not depend not only on fa Ca,a(E) and 0 < c < 1, but also on h.From inequality (0.42) it follows that 0.4.The Content of Paper This paper is devoted to investigation of well- posedness of differential equation -d2v/dt 2 + Av-f (-oc < < +oc) (0.38) and its difference analog in the arbitrary Banach space E.Here A is the (unbounded) closed linear operator in E with dense domain D(A).Equation (0.38) is con- sidered in the functional (abstract) H61der space Ca(E) (0 < c < 1), and for any positive (see [2]) Avhl 0() <-M.In 1/h Ilfhllc,,() (0 < h < !) 2 (0.43) Here C a (E) is the Banach space of uni- formly bounded grid functions /)h__ ()i E; i=-oc, +oc).Inequality (0.43) leads us to for- mula (0.23) of exact value M.(h) in difference coercive inequality (0.17).
To difference equation (0.39) is established for its solution v in Ca(E) with some I_<M< /oc, does not depend on f<=_ Ca(E) and 0 < c < 1.To differential equation (0.38) Grisvard's theory (see [3]) is applicable, but it leads us to the coercive inequality (0 < < 1/21 (0.41) Difference equation (0.39) is considered as the operator equation in the H61der space C h' (E) (0 < c < 1) of (abstract) grid functions, and for any strongly positive (see [2]) in E operator A coercive inequality IkAv (0.42) We will say that Eq. (1.1) is well-posed in C(E), if the following two conditions are fulfilled: (al) For any f C(E) there exists the unique solution v(t)= v(t; f) in C(E) of Eq. (1.1).It, in particular, means that d2{v(t;f)]/(dt) 2 and Av(t;f) (1.4)   are acting in C(E) additive and homogeneous operators, defining on whole Banach space C(E).(a2 inequality bounded in E inverse for any A >_ 0, and estimate (1.9) is true for some <M < +ec.Such A is called positive in E operator (see [2]).So, if Eq. (1.1) is well-posed in the functional Banach space C(E), then A is positive operator in the Banach space E (under condition that operator A -1 is bounded in E).Whether the positivity of operator A in E is sufficient condition of the well-posedness of Eq. (1.1) in C(E)?.
For arbitrary Banach space F let us consider the acting in C(F)= C[(-oc,+oc), F] operator A, defining by formula Ab(x) -b"(x) + b(x)(-oc, (1.10) Properties (al) and (a2), in virtue of Banach's theorem, lead us to coercive inequality IIv'llc(/+ IIAv(t)llc(l Me" Ilfllc(/ Inequality (1.6) permits to investigate the spec- tral properties of operator coefficient A for well- posed in C(E) of Eq. (1.1).For any u D(A) and A > 0 we will put on functions (x) C(F), such that "(x) C(F).
b Au + Au.
b(A) i (1.16) In particular, the negative fraction powers A (c>0) of positive operator A are defined (see [2]), A-C-(A-1) for integer c, and semigroup identity A -(+/) A -".A -(0 < a,/3 < +oc) (1.17) is true.From these statements, evidently, it fol- lows that positive fractional powers A(c >0) can be defined by formula A-(A-) -' (1.18) holds.Acting in the Banach space E linear opera- tor B with dense domain D(B) is called strongly positive (see [2]), if operator hi+ B has bounded inverse for any complex number A with Re A _> 0, and estimate (1.23) is true for some l_<M< +oo.Operator B is strongly positive iff-B is the generator of analy- tic semigroup exp{-tB} (t > 0) of linear bounded in E operators with exponentially decreasing norm, when + /oc, i.e. estimates exp{-tB} IIF+ E, tB exp{-tB} IIE-E <_ M(B) .e -a(B)t (t > O) (1.24) Operators A(c > 0) already are unbounded, and their domains D(A) are dense in E. The follow- ing moment inequality are true for some I<_M(B)< +ec, O< a(B) < +oc.Thus, is strongly positive in E operator, i.e. the following estimates hold: AulIEM(c,).IAuI/ Ilulle [0 < c < fl < +oc, u D(A)] exp{-tx/-} IIE_ , tV/--'exp{--tV/--}E_E _< e (t > 0). (1.25) The consideration of operator permits to re- duce differential equation (1.1) of the second or- der to equivalent system '() + ,/5. () (),'(t) + v. z(t) f(t) (-oc < < +oc) (1.26) of differential equations of the first order.This fact prompts that for solution v(t) in C(E) of Eq. (1.1) formula is true for its solution v(t) in Ca(E) with some l_<M(c0<+oc, does not depend on f(t) Ca(E).As in the case of space C(E) it is estab- lished that from coercive inequality (1.30) the positivity of operator A in Banach space E fol- lows.It turns out that this property of operator A in E is not only necessary, but also sufficient condition of well-posedness of Eq. (1.1) in Ca(E) for all c(0,1).(1.28) It turns out that formula (1.27) defines the unique solution in C(E) of Eq. (1.1) under essen- tially less restrictions on the smoothness of func- tion f(t).
Difference equation of the second order (2.1) is equivalent to system of difference equations of the first order defining by formula Avi-B.(2 + B) -1.
However under investigation of convergence of difference method it is necessary to establish the well-posedness of Eq. (2.1) in Banach space Ch(E) not for some fixed h(0,1) but in the aggregate of such spaces for all h (0,1].
The property (2.25) is called the stability of dif- ference equation (2.11) in the Banach space (2.29) are true for some < Ms(cO, Mc(c)< +oc, do to depend on f Ch'(E) and h (0, 1].
(2.39) true also in the case m 1.These estimates are analogous to estimates of analytic semigroup.They permit to establish the following result.
THEOaEM 2.1 Let A be strongly positive operator in the Banach space E. Then difference equation (2.1) is well-posed in the aggregate of Banach space Ch'(E (0 < c < 1) and coercive inequality and obtain Av Ich,() -< M-o -. (1 a) -.
It is easy to see that estimates (2.38) and (2.41) are