PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY

Transcription

1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Matheatical Sciences 13,, M a t h e a t i c s ON BOUNDEDNESS OF A CLASS OF FIRST ORDER LINEAR DIFFERENTIAL OPERATORS IN THE SPACE OF n 1)-DIMENSIONALLY CONTINUOUS FUNCTIONS V. Zh. DUMANYAN Chair of Nuerical Analysis and Matheatical Modeling YSU, Arenia In this article we consider first order linear differential oerators in the sace of n 1)-diensionally continuous functions with coefficients having soe growth near the doain boundary and rove the boundedness of considered oerators. MSC1: 3H5; 46E15. Keywords: bounded differential oerator, first order differential oerator, n 1)-diensionally continuous function. Introduction. Let R n, n, be a bounded doain with sooth boundary C 1. We introduce the following sace: U) u W,1oc) 1 C n 1 ) : rx) ux) dx <, u U) = u C n 1 ) + rx) ux) dx, where rx) is the distance fro a oint x to the boundary and C n 1 ) is the Banach sace of n 1)-diensionally continuous functions. The concet of n 1)-diensional continuity was introduced by Gushchin [1] and is as follows. Let µ be a nonnegative unit easure on R n, with suort in, and satisfying the following condition: there exists a constant C = Cµ) such that for all r > and x the easure of the ball B r x ) of centre x and radius r is ajorized by Cr n 1, i.e. µb r x )) Cr n 1 for all r > and x 1) E-ail: duan@ysu.a

2 Duanyan V. Zh. On Boundedness of a Class of First Order Linear Diff. Oerators... 9 the sallest such C is called the nor of µ, written µ. The Gushchin sace of n 1)-diensionally continuous functions C n 1 ) is the coletion of the sace of continuous functions on with resect to the nor u Cn 1 ) = su 1 u x)dµ x), µ where the sureu is taken over all easures µ satisfying 1). Functions in C n 1 ) L ) have traces on sets of ositive n 1)-diensional Hausdorff easure, and the set of traces of all functions fro C n 1 ) on the sooth n 1)- diensional surface Γ coincides with L Γ). There is an equivalent definition of n 1)-diensional continuity in ters of the roxiity of the values of functions on neighbouring easures. Let µ, ν be nonnegative unit easures on R n, with suort in, satisfying 1), and let φ be a easure on R n, with suort in. We also assue that µg) = φg R n ),νg) = φr n G) for all Borel) sets G. A function u is said to be n 1)-diensionally continuous, if for any ositive ε there exists a ositive δ such that 1 dφ ux) uy)) x,y) < ε µ + ν the distance between the values of u on the easures µ and ν along φ is less than ε), rovided x y dφ x,y) < δ the distance between the easures µ and ν along φ is less than δ); see [1] for ore details. Note that the sace U) was introduced in [1], in the aer devoted to the investigation of Dirichlet roble for second order ellitic equation without lower order ters. The study of Dirichlet roble for general second order ellitic equation requires, in articular, investigation of roerties of the first order differential oerator Tu b, u) + div cu) du, u U), where the coefficients bx) = b 1 x),..., b n x)), cx) = c 1 x),..., c n x)) and dx) are easurable and bounded on any strictly interior subdoain of. For an arbitrary u U), define a linear functional Tu acting on W 1 ) as Tu,v bx), ux))vx) cx)ux), vx)) dx)ux)vx))dx, v W 1 ).

3 1 Proc. of the Yerevan State Univ. Phys. and Math. Sci., 13,, The ain result of the resent aer is the following T h e o r e. Suose B t) dt <, where Bt) su bx), ) rx) t C t) dt <, where C t) su cx), 3) rx) t t D t) dt <, where Dt) su d x). 4) rx) t Then T is a bounded linear oerator fro U) to W 1 ). Proof. Let x be an arbitrary boundary oint of the doain. We fix a local coordinate syste x,x n ) with origin at x, and take the x n -axis along the inward noral vector νx ) to at x. The boundary being C 1 -sooth, there exists a nuber r x > and a function ϕ x C 1 R n 1 ), ϕ x ) =, ϕ x ) =, ϕ x x ) 1 for all x R n 1, such that the intersection of with the oen ball U r x ) x : x x < r x with x = centre x and radius r x, has the reresentation U r x ) = U r x x ) x,x n ) : x n > ϕ x x ). Hence, x U r x ) = U r x x ) x x,x n ) : x n = ϕ x x ). Let l x = r x. Choose U l x ) x, = 1,...,, to be a finite oen refineent of the oen cover,x of the boundary. Also let U l x ) x U = U r x ) x, r = r x, l = l x, ϕ = ϕ x, where = 1,...,. We set h = 1 ) inr 1,...,r ). Then each of the curvilinear cylinders 3 5 Π l,h = x,x n ) : x < l,ϕ x ) < x n < ϕ x ) + h, = 1,...,, lies in the corresonding ball U and hence in U. Let l < h be a ositive nuber such that the coleent in of the doain l = x : rx) = dist x, ) > l is contained in the union of the cylinders Π l,h, = 1,...,, that is, l = x : rx) = dist x, ) l Π l,h.

4 Duanyan V. Zh. On Boundedness of a Class of First Order Linear Diff. Oerators It can be easily seen that for any x = x,x n ) Π l,h, = 1,...,, 5 rx) x n ϕ x ) rx). We fix soe, 1, and consider a local coordinate syste with origin at x. In what follows, we suress the deendence of the function ϕ on, writing for brevity ϕ = ϕ. We define the aings L and L 1 of the sace R n onto itself by Lx) = x,x n ϕx )), x = x,x n ), L 1 y) = y,y n + ϕy )), y = y,y n ), resectively. Let Π l,h = LΠ l,h ) be the iage of Π l,h under L. Given arbitrary u U) and η C ), we set uy,y n + ϕy )) = ũy), ηy,y n + ϕy )) = ηy). Further, consider Tu,η bx), ux))ηx)dx cx)ux), ηx))dx dx)ux)ηx)dx. ) 1 First, it is readily verified that f t) = o at t + for a onotonously t decreasing function f t), t, having the finite integral f t)dt <. Then, using ), we obtain Bt) const t and bx), ux))ηx) dx bx) ux) ηx) dx rx) ux) dx const u U) Further, by Hardy s inequality [4], η x) r x) dx l B rx)) rx) η x) r x) dx η x) r x) dx + Π l,h η x)dx. η x) r x) dx 1 l η x)dx + ) 5 Π l,h η y) y dy = 1 n l η x)dx+

5 1 Proc. of the Yerevan State Univ. Phys. and Math. Sci., 13,, yn ) ) h η τ y,τ)dτ 5 o + dy n y dy const η dx+ y n <l ) h y n 5 + dy η τ y,τ)dτ/y n dy n y <l o const η h dx + dy η y n y,y n )dy n y <l const η dx + ηy) dy const ηx) dx. 5) Hence, Π l,h bx), ux))ηx) dx const u U) ηx) dx where the constant is indeendent of u and η. const u U) W 1 ), 6) Using 3), we obtain cx)ux), ηx))dx Crx)) ux) ηx) dx C rx))u x)dx ηx) dx const C rx))u x)dx const C l ) u x)dx + l const u L ) + Π l,h Π l,h W 1 ) C rx))u x)dx C 5 y n )ũ y,y n )dy dy n W 1 ) W 1 )

6 Duanyan V. Zh. On Boundedness of a Class of First Order Linear Diff. Oerators const u L ) + h C 5 y n )dy n ax y n h y <l const u L ) + ax ũ y,y n )dy y n h y <l ũ y,y n )dy const u L ) + u C n 1 )) W 1. ) Further, it follows by C n 1 ) L ) that cx)ux), ηx))dx const u C n 1 ) W 1 ) W 1 ) W 1 ) const u U) W 1 ), 7) where the constant is indeendent of u and η. Alying 4) and 5), we have, by analogy with the revious estiates, dx)ux)ηx) dx Drx)) ux) ηx) dx const + D l ) Π l,h r x)d rx))u x)dx const r x)d rx))u x)dx l r x)u x)dx + const Π l,h D l ) ax x l r x) η x) r x) dx W 1 ) r x)d rx))u x)dx 5 y nd y n )ũ y,y n )dy dy n l u x)dx+ W 1 ) W 1 ) const u L ) +

7 14 Proc. of the Yerevan State Univ. Phys. and Math. Sci., 13,, h y n D 5 y n ) dy n ax const u L ) + ax const y n h y n h ũ y,y n )dy y <l y < l ũ y,y n )dy W 1 ) W 1 ) u L ) + u C n 1 )) W 1 ) const u U) W 1 ), 8) where the constant is indeendent of u and η. Consequently, it follows fro 6), 7) and 8) that for arbitrary u U) and η C ) Tu,η const u U) W 1 ), where the constant is indeendent of u and η. Since the functions η of C ) for a dense subset of W 1 ), the estiate just obtained forces the oerator T : U) W 1 ) to be bounded. Reark. It is clear, that condition ) of the Theore can be relaced by the following: bx) const r x), x. Received R E F E R E N C E S 1. Gushchin A.K. On the Dirichlet Proble for a Second-Order Ellitic Equation. // Mat. Sb. N.S.), 1988, v ), 19), Duanyan V.Zh. On Solvability of the Dirichlet Proble for Ellitic Equations. // Izvestia NAN Arenii. Mateatika, 11, v. 46,, in Russian). 3. Duanyan V.Zh. Solvability of the Dirichlet Proble for a General Second-Order Ellitic Equation. // Mat. Sb., 11, v., 7, in Russian). 4. Maz ya V.G. Sobolev Saces. L.: LGU, 1985, 416. in Russian).